Even though it is less general than some other definitions, the Riemann sum definition is very close to the way that integration is interpreted and used in geometry and in applied mathematics. When I set up an integral, say to find the volume of a domain in $\mathbb{R}^n$ or the volume of a manifold, I basically derive the integrand as an idealized (possibly multidimensional) Riemann sum.
For example, say that I want to find the integral of a function $f(\theta,\phi)$ on the sphere $S^2$. Then the limit of a Riemann term is the value of $f$ times a little region in the shape and orientation of Colorado (as if the sphere were the Earth) that subtends angles of $d\theta$ and $d\phi$. The region has height $d\theta$ and width $(\sin \theta)d \phi$, hence the integral is $$\int_{S^2} f(\theta,\phi) (\sin \theta) d\phi d\theta.$$ Admittedly this informal model can be adapted to Lebesgue integration as well as to Riemann integration --- but not especially to Cauchy's trick that Dieudonné promotes. So, Lebesgue integration is a good thing to teach, but it is clearly more complicated than Riemann integration. It's really an upper-division undergraduate topic, or a first-year graduate topic, to explain why Lebesgue integration is an excellent definition and not a gratuitously complicated one. (But it makes sense, when you teach Riemann integration, to briefly state that Lebesgue integration cures particular diseases of Riemann integration.)
