Here are two approaches:

1. The probability that your first roll will come up with some number 
is 1, for it's certain.

The probability that the second roll will not match whatever number you 
rolled first is 5/6.

The probability that the third roll will not match either the number you 
rolled first or the number you rolled second is 4/6.

Answer: 1*(5/6)*(4/6) = 20/36 = 5/9


Another way to do it:

2. There are 6*6*6 = 216 ways the three dice could come up.
There are 6P3 = 6*5*4 = 120 ways they could all three be different.

The probability is 120 ways out of 216 or 120/216 which reduces
to  5/9.

Edwin