From a conceptual standpoint, I think that there are three things one asks of an approach to integration
1) An easily accessible geometric interpretation
2) A readily available computational toolbox (e.g. the fundamental theorem of calculus)
3) A flexible theory
The Lebesgue integral is absolutely unrivaled in (3), but it is actually quite obtuse from the other two points of view. Basic results like the Lebesgue differentiation theorem and the change of variables formula are not at all transparent from the Lebesgue point of view, and geometrically it is no better than the Riemann integral. The Cauchy integral is great if you only care about (2), but it is abysmal at (1) and (3). The Riemann integral, for all its faults, strikes a pretty good balance between (1) and (2). It is even known to enjoy an occasional technical advantage over the Lebesgue theory; for instance, one must invent the theory of distributions to make sense of the Cauchy principal value of an improper integral in the Lebesgue theory if I recall correctly.
In line with what others have said in the comments, which of the three criteria you care most about really depends on the audience.
For a class full of engineers and business majors, the question is essentially moot: two students out of a hundred can correctly define the integral of a continuous function at the end of the semester. But in my view the real point of such classes is to help students develop the language and the skills necessary to reason with rates of change. So in the end it hardly matters what the precise definitions were, and to the extent that it does matter the Riemann integral is quite well suited.
For a class full of grad students, on the other hand, the point is to show the students how to prove theorems. So there is no choice but to use the Lebesgue integral, and the Riemann integral can be largely ignored (as it is in most graduate real analysis classes).
The gray area lies in more advanced undergraduate analysis classes. Often these classes are populated by math, physics, and engineering majors who intend to actually do something with mathematics one day. The problem with teaching the Lebesgue integral to such students is that with undergrads you have to spend half the semester on measure theory, detracting from the time that should be spent on the topology of Euclidean space, multilinear algebra, and whatever else belongs in such a course. I can't even imagine how one builds integration in higher dimensions from the Cauchy point of view, short of turning Fubini's theorem into a definition. So the Riemann integral seems like a perfectly reasonable compromise to me. I admit that I found it a little frustrating to learn a theory that I even knew at the time I was probably never going to use, but I lived through the experience.
