What's the probability of x dice, each with y sides, adding up to z?

Picture X fair dice, each of which has Y sides, labeled from 1 to Y. How likely is it that when the dice are rolled, the sum of the showing sides will equal Z, for some Z between X and X * Y?

Of course, one could figure this out by generating all the possible combinations of the X dice and then seeing which ones add up to Z, but in a universe that has given such wonders as penicillin, space travel, and Lucy Pinder, I have to believe there's a smarter way to do it.