jmc

First, we choose the suits. There are $\left(\genfrac{}{}{0ex}{}{4}{3}\right)=4$ ways to do this. Then, we choose one of 13 cards from each of the chosen suits. There are $13^3=2197$ ways to do this. The total number of ways to choose 3 cards of different suits is therefore $4\cdot 2197=\boxed{8788}$.