I have found most theoretical presentations of integration, whether Riemann, Lebesgue, or something else, to be overkill. I believe every mathematician should know how to construct the Riemann and Lebesgue integrals and have a solid understanding of their properties, but both can almost always be used as black boxes, when proving theorems in, say, PDE's and differential geometry. I in fact try to use only the 1-d Riemann integral as much as possible and get away with this surprisingly often. More sophisticated ideas should be brought in only if absolutely needed.
ADDED: Let me add that I believe strongly that undergraduate and graduate math students (including at the Ph.D. level) should be trained not just in the theoretical aspects of math but also in using math as a practical tool (especially given how many do not end up as research mathematicians). The concept of a Riemann integral and how to approximate it is arguably one of the most useful in mathematics. So although my comment above focused on the usual suspects (engineers), it applies to mathematicians as well.