In the introduction to chapter VIII of Dieudonné's Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument: 


  Finally, the reader will probably observe the conspicuous absence of
  a time-honored topic in calculus courses, the “Riemann integral”. It may
  well be suspected that, had it not been for its prestigious name, this would
  have been dropped long ago, for (with due reverence to Riemann’s genius)
  it is certainly quite clear to any working mathematician that nowadays such
  a “theory” has at best the importance of a mildly interesting exercise in the
  general theory of measure and integration (see Section 13.9, Problem 7).
  Only the stubborn conservatism of academic tradition could freeze it into
  a regular part of the curriculum, long after it had outlived its historical
  importance. Of course, it is perfectly feasible to limit the integration process
  to a category of functions which is large enough for all purposes of
  elementary analysis (at the level of this first volume), but close enough to
  the continuous functions to dispense with any consideration drawn from
  measure theory; this is what we have done by defining only the integral
  of regulated functions (sometimes called the “Cauchy integral”). When
  one needs a more powerful tool, there is no point in stopping halfway, and
  the general theory of (“Lebesgue”) integration (Chapter XIII) is the only
  sensible answer.


I've always doubted the value of the theory of Riemann integration in this day and age.  The so-called Cauchy integral is, as Dieudonné suggests, substantially easier to define (and prove the standard theorems about), and can also integrate essentially every function that we might want in a first semester analysis/honors calculus course.  

For any other sort of application of integration theory, it becomes more and more worthwhile to develop the fully theory of measure and integration (this is exactly what we did in my second (roughly) course on analysis, so wasn't the time spent on the Riemann integral wasted?).  

Why bother dealing with the Riemann (or Darboux or any other variation) integral in the face of Dieudonné's argument?

Edit: The Cauchy integral is defined as follows:

Let $f$ be a mapping of an interval $I \subset \mathbf{R}$ into a Banach space $F$. We
say that a continuous mapping $g$ of $I$ into $F$ is a primitive of $f$ in $I$ if there
exists a denumerable set $D \subset I$ such that, for any $\xi \in I - D$, $g$ is differentiable
at $\xi$ and $g'(\xi) =f(\xi)$ .

If $g$ is any primitive of a regulated function $f$, the difference $g(\beta) - g(\alpha)$,
for any two points of $I$, is independent of the particular primitive $g$ which
is considered, owing to (8.7.1); it is written $\int_\alpha^\beta f(x) dx$, and called the integral
of $f$ between $\alpha$ and $\beta$. (A map $f$ is called regulated provided that there exist one-sided limits at every point of $I$).  



Edit 2: I thought this was clear, but I meant this in the context of a course where the theory behind the integral is actually discussed.  I do not think that an engineer actually has to understand the formal theory of Riemann integration in his day-to-day use of it, so I feel that the objections below are absolutely beside the point.  This question is then, of course, in the context of an "honors calculus" or "calculus for math majors" course.


  It is frequently claimed that Lebesgue integration is as easy 
  to teach as Riemann integration. This is probably true, but I 
  have yet to be convinced that it is as easy to learn.


T. Körner: A companion to analysis: a second first and first second course in analysis.

I know, this is slightly besides the Riemann vs Cauchy point, but I like this quotation so much I couldn't help myself...
