Is Euclid dead? Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (see King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, p. 157), often associated in the popular mind with Bourbaki's general stance on rigorous, formalized mathematics (eschewing pictorial representations, etc.). See Dieudonné's address at the Royaumont seminar for his own articulated stance.
In brief, the suggestion was to replace Euclidean Geometry (EG) in the secondary school curriculum with more modern mathematical areas, as for example Set Theory, Abstract Algebra and (soft) Analysis. These ideas were influential, and Euclidean Geometry was gradually demoted in French secondary school education. Not totally abolished though: it is still a part of the syllabus, but without the difficult and interesting proofs and the axiomatic foundation. Analogous demotion/abolition of EG took place in most European countries during the 70s and 80s, especially in the Western European ones. (An exception is Russia!) And together with EG there was a gradual disappearance of mathematical proofs from the high school syllabus, in most European countries; the trouble being (as I understand it) that most of the proofs and notions of modern mathematical areas which replaced EG either required maturity or were not sufficiently interesting to students, and gradually most of such proofs were abandoned. About ten years later, there were general calls that geometry return, as the introduction of the alternative mathematical areas did not produce the desired results. Thus EG came back, but not in its original form.
I teach in a University (not a high school), and we keep introducing new introductory courses, for math majors, as our new students do not know what a proof is. [Cf. the rise of university courses in the US that come under the heading "Introduction to Mathematical Proofs" and the like.]
I am interested in hearing arguments both for and against the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs?  If so, is there a more efficient mathematical subject in comparison to EG, for high school students, in order to learn what is a theorem, an axiom and a proof?
Full disclosure: currently I am leading a campaign for the return of EG to the syllabus of the high schools of my country (Cyprus). However, I am genuinely interested in hearing arguments both pro and con.
 A: When I was in high school (in the early 1960's), Euclidean geometry was the only course in the standard curriculum that required us to write proofs.  These proofs, however, were in a very rigid format, with statements on the left side of the page and a reason for every statement on the right side.  So I fear that many students got an inaccurate idea of what proofs are really like. They also got the idea that proofs are only for geometry; subsequent courses (in the regular curriculum, not honors courses) didn't involve proofs.  The textbook that we used also had some defects concerning proofs.  For example, Theorem 1 was word-for-word identical with Postulate 19; Theorem 1 was given a proof that didn't involve Postulate 19, so, in effect, we were shown that Postulate 19 is redundant, but the redundancy was never mentioned, and I still don't know why a redundant postulate was included in the first place.  Another defect of the standard courses in geometry was that, because of the need to gently teach how to find and write proofs (in that rigid format), very little interesting geometry was taught; the class was mostly proving trivialities.  I was fortunate to be in an honors class, with an excellent instructor who showed us some really interesting things (like the theorems of Ceva and Menelaus), but most students at my school had no such advantage.
I conjecture that Euclidean geometry can be used for a good introduction to mathematical proof, but, as the preceding paragraph shows, there are many things that can go wrong. (There are other things that can go wrong too.  I mentioned that I had an excellent teacher.  But my school also had math teachers who knew very little about proofs or about geometry beyond what was in the textbook.)  So my advice is, if you want to develop a course such as you described in the question, proceed, but be very careful.
Incidentally, many years ago, I recommended to my university department that we use a course on projective geometry as an "introduction to proof" course.  The idea was that there are fairly easy proofs, and the results are not as obvious, intuitively, as equally easy results of Euclidean geometry.  My suggestion was not adopted.  
Qiaochu Yuan's suggestion of discrete math instead of geometry might have similar advantages as my projective geometry proposal, but it will still be subject to many of the pitfalls that I indicated above, plus one more: Most high school math teachers know less about discrete math than they do about geometry.
A: I try to keep my answer short.
Fact: Euclidean geometry is still taught in Iranian middle and high schools.
Observation (based on research): Most teachers do not like to teach geometry. They say, when you teach geometry, you are always faced with problems that you don't know how to solve. But, it seems that they haven't got that problem with the rest of mathematics taught in school! Thinking of your campaign, ask yourself, have you got enough teachers willing to teach geometry and able to do so?
Fact : There is at least one mathematician who is in love with triangles. Here is a quote from his paper The Mathematics of Mathematics Houses (The Snaky Connection) in The Mathematical Intelligencer:

No object has ever served mathematics better or longer. Compare the
number of nontrivial results which are true for all topological
spaces, rings, groups, etc, without putting extra assumptions on them
with the number of nontrivial results which are true in any
triangle. … When it comes to deducing results in mathematics just from
the definition of an object, nothing can hold a candle to the
triangle. The triangle will serve mathematics forever.

Opinion: There is a big difference between teaching geometry as a source of fascinating problems, and as a rigid body of axiomatic knowledge. Personally, I favor the former. Go to observation above!
A: As long as this question is open I might as well throw in my two cents. I think it is not useful to teach Euclidean geometry to high school students. Here are some reasons I can think of for people to teach Euclidean geometry to high school students and why I think they are bad reasons:


*

*As an introduction to the notion of a proof. As I said in the comments, I think there are better options here, such as areas of discrete math like elementary number theory, elementary combinatorics, or elementary graph theory. Unlike Euclidean geometry, at least some of this material has nontrivial applications: for example, the application of elementary number theory to cryptography or the application of combinatorics to analyzing algorithms. Also unlike Euclidean geometry, this material offers a lot of opportunity for computer-based exploration: for example, Project Euler. But it's not even clear to me that high school students really need an introduction to proof. 

*As preparation for other topics that high school students ought to know. Euclidean geometry might not be a bad way to prepare students for trigonometry and eventually calculus, but I don't think high school students ought to learn these things either. The same goes for physics. 

*As preparation for using mathematics in daily life. Here I think topics like Fermi estimation and some basic probability and statistics would be more useful (e.g. for helping people make better political and medical decisions). As far as I can tell most people have no use for Euclidean geometry in their daily lives. 

*As preparation for jobs involving mathematics. If students want to take such jobs, the relevant mathematics can be taught to them as part of their job training, or they can pick it up themselves. Note that there are many people with programming jobs despite the general lack of programming in most high school curricula. 
A: I strongly recommend to read the paper Нужна ли школе 21-го века Геометрия? of Sharygin.
(It is in Russian, but it worth to translate.)
You will see the reasons to return EG in school, you will also the reasons why it disappears.
Sharygin is my hero, he is the author of many very good math books for school students, he also wrote the best (the opinion is mine) text book in Euclidean geometry for school.
P.S. Let me share what I know about the history of geometry curriculum  in  Russian school. We had textbook of Kiselev, which served for more than half century. It was changing slowly, at the beginning it was quite close to Euclid's Elements. (If you ask about geometry someone from the generation of my parents, their eyes start to radiate with positive energy and they start to explain how wonderful was the experience.)
After that (60-s) changes start. First Nikitin's book — a big step back. After that, instead of coming back to Kiselev, many books were written by very prominent mathematicians (including Alexandrov and Pogorelov); these books were yet worse than Nikitin's book. Later Sharygin's book appears; it is a very good book but extremely demanding from the teacher (say absolute geometry was not discussed, but if the teacher is not familiar with absolute geometry then he can not teach properly).
Now we get so called "Unified state examination" (the worst reform ever made in Russia); it is either too expansive or impossible to check proofs on this exam; the later wipes geometry from the school curriculum; formally it is still there but since it is not needed to pass the exam, no one needs to learn it.
Conclusion: It seems that every big reform makes education worse. The right direction would be to change things gradually, and it has to be done by teachers with help of academia, not other way around.
A: Euclidean geometry is still taught in American high schools, but I am strongly against it.  I think it should be replaced with linear algebra.  
Arguments against Euclidean geometry:


*

*Most of what you prove in a high school Euclidean geometry class seems pretty obvious until you learn about non-Euclidean geometry.  It makes students think that proofs are pedantry for its own sake.

*Euclidean geometry is basically useless.  There was undoubtedly a time when people used ruler and compass constructions in architecture or design, but that time is long gone.  

*Euclidean geometry is obsolete.  Even those students who go into mathematics will probably never use it again.
Arguments for linear algebra:


*

*$\mathbb{R}^2$ with the standard inner product is a model for the Euclidean axioms, so in particular you can still prove the same theorems if you really want to.

*Linear algebra generalizes easily to dimensions larger than 3 where most students' geometric intuition breaks down, so it is easier for them to appreciate the need for axioms and theorems.

*Linear algebra - particularly eigenvalues and eigenvectors - is ubiquitous in modern science and engineering.  I would argue that the average person is much more likely to encounter an eigenvalue problem than a calculus problem.

*Linear algebra is, of course, still the basic language in which most of mathematics is expressed and thus a linear algebra class is a more honest taste of what math is all about.  

*Providing students with an early foundation in linear algebra would make later education run more smoothly.  Even many non-scientists use software that is based on solving linear systems or computing matrix decompositions, and it might help for such people to have a little more context.  And those who go on to take further science classes - particularly physics - would more obviously benefit.  If nothing else, we might finally be able to teach our students the correct second derivative test in multivariable calculus classes...
A: With the caveats mentioned by Andreas, I think Euclidean Geometry makes excellent sense as a high-school course. (My high school experience was not dissimilar to Andreas's -- still the two-column format, but I also had a teacher who understood mathematics beyond what was in the textbook.) 
The basic point of agreement (between those Bourbakistes and those who would uphold EG) seems to be that there is need for a course that expounds mathematics as an axiomatic discipline, and the careful modes of reasoning that go into that. In some sense just about any system based on axioms (be it EG, set theory, "discrete mathematics", or something else) would serve that purpose, except that Euclidean Geometry has the big advantage of being visual and readily accessible to intuition. 
(The downside to that might be Isaac Newton's criticism [see Arnold's Huygens and Barrow, Newton and Hooke, pp. 49-50] that most of the theorems are intuitively quite obvious, so that the typical course can seem a painful exercise in pedantry.) 
I like Andreas's projective geometry proposal. Among other things, this would help promote the idea of the power of unification in mathematics: that things that might look very different, such as ellipses and hyperbolas, are often the same thing in disguise. 
A: There is something important besides rigor introduced in Euclidean Geometry classes: a connection between visual perception and sequential reasoning. 
In "Mathematics in the 20th Century" Atiyah likened Geometry with space-bound visual perception and Algebra with sequential time-bound reasoning. If we continue that simile a course that naturally combines both would be a movie, something much more than the ingredients. And every time we encounter one of those movies it usually generates quite a bit of excitement.
Algebra, however, is not the only sequential process in Mathematics; the other one is the sequential reasoning of a proof.
What I find important EG is that it's the first course in High School that connects visual perception and sequential reasoning, making it the first "movie" the kids ever see, and for many of them the only one. Replacing EG with Number Theory or Combinatorics as other suggested would replace the marriage of visual to sequential with a marriage of sequential to sequential.
A: Also in Israel, Euclidean geometry is taught in schools for quite some time (judging from my parents, me, and my children). I personally like the idea of it being taught and being the first encounter with mathematical axioms, definitions and proofs, as well as an encounter with geometrical thinking. For learning what a mathematical proof is, I doubt if any of the suggested substitutes will even come close. 
But it is not clear to me how crucial it is to teach (everybody) the notion of a mathematical proof in high school at all. 
A: It might be interesting to point out that the support for  removing axiomatically taught Euclidean geometry (not all geometry) from the school education predates Bourbaki. Oliver Heaviside (1850-1925), a British mathematician who also made important contributions to physics wrote in his "Electro-Magnetic Theory", vol. 1 (1893):
"As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous".
A: I completely agree with you.  It is important for everyone to be exposed to proofs, because it shows them what math is really about--reasoning, not computation.  The mathematical way of thinking is very valuable for developing general critical thinking skills and the most careful and precise reasoning.  I believe there is no substitute.  You also hit the nail on the head when you point out that Euclidian geometry is a great medium for learning what proofs are all about.  The subject matter connects with intuition, and the propositions and arguments are easily seen to be well-motivated and accessible to the novice.  As you mention, it is empirically found to be hard to do proofs for the beginner with other topics.
How can we expect our undergraduates to do well when the secondary education is lacking in the prerequisite training?  If mathematics education is an appropriate topic of discussion here, then certainly the relation of high school curriculum to the preparedness of undergraduates is relevant.
A: You need to establish a goal, and the reason for the goal.  An example is "Have my high school require everyone to take a course covering this syllabus in Euclidean geometry" for the goal, and "because it is intellectually enriching and potentially useful" as the reason.
I don't think the above is a good example.  Here is a different example: "Require knowledge of Euclidean geometry and its applications to graduate from high school" as a goal, with the reason being "our society needs engineers, technicians, and other workers who will use the knowledge and applications to improve our community."  I like this example a little better because the reason feels more concrete; sadly, I do not know if the reason is valid.
As your present question stands, I do not see a good combination of goal and reason.  When you have that, you will have a foundation for arguing for your goal. 
If the goal is to help students learn proofs, I might suggest looking at Common Core education standards happening in the United States.  Good communication and expression in a broad range of areas of study is emphasized, and I would couple this with the ability to produce arguments in a variety of styles: logical, emotional, inspirational, to start.
I would suggest a course or two which presents arguments in geometry, algebra, analysis, discrete mathematics, and logic, so that one can taste the different flavors of proof that occur in the fields.
Gerhard "Also Gives Fresh, Minty Breath" Paseman, 2013.12.19 
A: I'm suggesting teaching the foundations of mathematics. I would pick a system like:

*

*ETCS (Elementary Theory of the Category of Sets),

*HoTT (Homotopy Type Theory),

*Dependent Type Theory,

*ZFC?

I suggest this can conclude with one of:

*

*The theory of cardinal numbers.

*Defining the arithmetic operations on $\mathbb N, \mathbb Q, \mathbb R, \mathbb C$ and proving their identities.

*I'm perhaps overly idealistic, but perhaps some funky topos theory: Nonstandard Analysis? Synthetic Differential Geometry? Synthetic Computability?

Advantages:

*

*They'll learn mathematical notation and terminology. People who don't learn this notation and terminology might otherwise struggle with more advanced maths. Examples of terms they'll learn: Functions, tuples, Cartesian product, sets, subsets, natural number, etc.


*They'll learn a proof calculus like Natural Deduction, along with proof by induction. These are idealised, general and rigorous models (imitations?) of the proofs constructed by actual human beings.


*Has some link to topics like Functional Programming.
Disadvantages:

*

*Not sure how this can help most engineers.


*No overlap with geometry.


*No help with calculus, except clarifying some basic terms.


*Could be accused of being pedantic.
Questions:

*

*Will students or teachers find this easy to learn or teach?


*Where are the problems to solve? ---- Here I suggest cardinal numbers might provide a small problem list.
