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.
Best Answer
From a conceptual standpoint, I think that there are three things one asks of an approach to integration
1) An easily accessible geometric interpretation
2) A readily available computational toolbox (e.g. the fundamental theorem of calculus)
3) A flexible theory
The Lebesgue integral is absolutely unrivaled in (3), but it is actually quite obtuse from the other two points of view. Basic results like the Lebesgue differentiation theorem and the change of variables formula are not at all transparent from the Lebesgue point of view, and geometrically it is no better than the Riemann integral. The Cauchy integral is great if you only care about (2), but it is abysmal at (1) and (3). The Riemann integral, for all its faults, strikes a pretty good balance between (1) and (2). It is even known to enjoy an occasional technical advantage over the Lebesgue theory; for instance, one must invent the theory of distributions to make sense of the Cauchy principal value of an improper integral in the Lebesgue theory if I recall correctly.
In line with what others have said in the comments, which of the three criteria you care most about really depends on the audience.
For a class full of engineers and business majors, the question is essentially moot: two students out of a hundred can correctly define the integral of a continuous function at the end of the semester. But in my view the real point of such classes is to help students develop the language and the skills necessary to reason with rates of change. So in the end it hardly matters what the precise definitions were, and to the extent that it does matter the Riemann integral is quite well suited.
For a class full of grad students, on the other hand, the point is to show the students how to prove theorems. So there is no choice but to use the Lebesgue integral, and the Riemann integral can be largely ignored (as it is in most graduate real analysis classes).
The gray area lies in more advanced undergraduate analysis classes. Often these classes are populated by math, physics, and engineering majors who intend to actually do something with mathematics one day. The problem with teaching the Lebesgue integral to such students is that with undergrads you have to spend half the semester on measure theory, detracting from the time that should be spent on the topology of Euclidean space, multilinear algebra, and whatever else belongs in such a course. I can't even imagine how one builds integration in higher dimensions from the Cauchy point of view, short of turning Fubini's theorem into a definition. So the Riemann integral seems like a perfectly reasonable compromise to me. I admit that I found it a little frustrating to learn a theory that I even knew at the time I was probably never going to use, but I lived through the experience.