I have trouble matching up your post title with the associated text. I don't have time to fully work this out right now, but here is the basic idea:
P(all 3 rolls different) = 1 - P(2 or more of the rolls are the same)
Call the three dice d1, d2, and d3 then we have:
P(2 or more of the rolls are the same) = P(d1=d2) + P(d1=d3) + P(d2=d3) - 2*P(d1=d2=d3)
(the last term corrects for multiple counting of d1=d2=d3 in the first three cases)
Then each of those terms can be expressed as the number of possible ways the case can occur times the probability of each. For example, the last term is (assuming 6 sided die):
P(d1=d2=d3) = 6 * (1/6)^3
Hopefully this is enough to get you started. If not, you might try working out a simpler case (say 3 x d3 or 2 x d6) and try to cast it in the form I described.