I haven't really thought this through, but how does one actually compute integrals? For the Riemann integral one can either prove the fundamental theorem and have a large table of derivatives handy, or one can rely on material which is already present in the calculus sequence (summing sequences) and which is independently useful; that alone, I think, makes it the more pedagogically sane choice. (Related to Pete Clark's answer, Putnam questions also have a tendency to hide Riemann integrals as sums.sums. Slightly more meaningfully, as Pete says in the comments, I can't imagine doing Lebesgue or Cauchy integrals on numerical data.)
For the Lebesgue integral it seems one proves the fundamental theorem or proves a comparison theorem to the Riemann integral (well, or both). For the former, you get essentially the same result (as far as the average student is concerned) with less pain from Riemann integration; for the latter, you have to know something about Riemann integrals anyway.