Definitely, the one I like the most is the proof via ultrafilters. You only have to state the compactness of a topological space in terms of ultrafilters, which is easily obtained by the definition via open coverings (warning: the equivalence of the definitions is where one uses AC)
X is compact if and only if every ultrafilter is convergent.
Then one observes that
any continuous image of an ultrafilter is an ultrafilter (in particular, any projection from a product space)
any filter in the product space converges if and only if all its projections converge .
You really only need a few definitions and few natural properties. My test about how nice is a proof is: can I teach it to somebody just while standing in the queue at the canteen, on into subway car?