As far as I know, there are four basic possibilities for defining the real numbers:
The real numbers can be defined using axioms.
The real numbers can be defined using decimal expansions.
The real numbers can be defined using Dedekind cuts
The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers.
Approach #1 is probably the easiest, but it is unsatisfying, because you don't actually prove that the real numbers exist. That is, you don't demonstrate the existence of any number system that satisfies the given axioms. I think #1 is probably the best if a course is short on time, or if the students aren't mathematically sophisticated enough to benefit from seeing one of the constructive definitions.
Approach #2 is what the students are expecting, but unfortunately the details are quite cumbersome, and the construction is not very natural (since it depends on a choice of base). It also doesn't give the students much new insight into the nature of the real numbers---it just fills in the details of insight they already have. In my opinion, the only reason to use #2 is if you are independently interested in talking about decimal expansions, e.g. if most of the students in the class are future high-school teachers.
Approach #3 is quite simple and elegant, and is certainly the easiest of the constructive approaches. It's wonderfully intuitive, and really cements in the students minds the idea that the real numbers are obtained by "filling in" holes in the rational number line. The disadvantage of this approach is that it's primarily order-theoretic: students are unlikely to see a construction like this again unless they study order theory, so it might be better to use an approach that's connected to other things the students will be learning.
Approach #4 is complicated, but has three main advantages (1) Students in a real analysis course should probably learn about Cauchy sequences anyway, so the extra time isn't exactly wasted (2) It starts the course off by identifying sequences as the important thing, a theme that is likely to continue for the rest of the course (3) It is a construction that can be generalized to completions of arbitrary metric spaces, which is something that students will need to understand when they learn point-set topology or advanced analysis. The disadvantage, of course, is that this is certainly the most complicated of the four options, and some students will be completely mystified.
My opinion is that the choice of approach should be based on the audience and the nature of the course. No matter what, I think it's worth pointing out that the other approaches exist, and perhaps briefly discussing them in class.
So here's what I think:
For a one-semester course in real analysis, it's probably best to stick with an axiomatic approach. There are more important topics to cover than constructions of the real numbers.
For a two-semester course in real analysis for typical math majors, it probably makes sense to use either Dedekind cuts or Cauchy sequences. For average students, I think Cauchy sequences are a bit too difficult, so I would start with Dedekind cuts.
For a two-semester course in real analysis for future mathematicians (i.e. student planning to go to Ph.D. programs), I would recommend covering both Cauchy sequences and Dedekind cuts, as well as mentioning the other two approaches. This audience will benefit a lot from the Cauchy sequences approach, but they should also know about other possible approaches, since they may be teaching real analysis themselves one day!
For an analysis course aimed at future high-school teachers, I would think it would be a good idea to discuss the decimal number system a bit at the beginning. Depending on the length of the course, it might also be helpful to discuss Dedekind cuts and/or the axiomatic approach.
One final note: if you do use the axiomatic approach, there is still some choice in which version of completeness you include as an axiom. The tradition seems to be to use the least upper bound property, but I think there is good reason to use either the convergence of Cauchy sequences or the nested closed intervals property instead. The advantages of using a Cauchy-sequence-based axiom should be obvious, while the advantages of the nested closed intervals property are that it doesn't require any preliminary definitions, it connects well to notions of compactness, and it is perhaps the easiest of the three to understand.