@Noam: The triangle $(x,0,0),(0,y,0),(0,0,z)$ ought to have rational area, for a "totally rational" tetrahedron. I checked the first two examples on the Wikipedia page for Euler bricks, and they failed this test. For example, the triangle with side lengths 125,244,267 does not have rational area.

@Qiaochu: The combinatorial Nullstellensatz does not imply the Schwartz-Zippel lemma. Ex. 9.1.1 in Tao and Vu's Additive Combinatorics does not ask to derive the Schwartz-Zippel from the CNS but merely to prove it by modifying the argument to the proof. (You have to turn the page ;-) see books.google.de/…

@Dylan: Schwartz-Zippel is a QUANTITATIVE statement, as opposed to the combinatorial Nullstellensatz, which is an EXISTENCE statement. When you read the proof in Schwartz' paper, you see that it implies that there are at least (|S1|-t1)(|S2|-t2)...(|Sn|-tn) nonzeros. (The t_i's are not the same as those in CNS; the assumptions on t_i's are not directly comparable; both conditions are subsumed by Michał Lasoń, A generalization of Combinatorial Nullstellensatz, The Electronic J. of Combinatorics (2010), Article no. #N32, 6 pp., mentioned in one of the answers, by "Seva".