jmc

Suppose each team played just one game with each of the remaining teams. Then each of the nine teams plays eight games. This makes a total of $9\times 8$ or 72 games. However each game has been counted twice in this total. For example, the game between Team A and Team B appears in A's 8 games and also B's 8 games. Therefore there are $9\times \frac{8}{2}=36$ different games played. Since each game is played three times, the total number of games played is $3\times 36=\boxed{108}$.