Yes, you can use Cantor's diagonal argument to enumerate all triples. No, it does not answer the question whether every integer can be written as the sum of three cubes; it only would if the set of integers was finite.
I had exactly this in mind (or the proof that the set of rational numbers are countable), but mistakenly thought it was called the 'diagonal argument' because you would get the bijection to the natural numbers by counting diagonally through the table.
