Steve,

Suppose the three teams are players 1, 2, 3, players 4, 5, 6, and players 7, 8, 9. There is a schedule where, over three days, any two players on different teams have played together exactly once:

Day 1: 1, 4, 7; 2, 5, 8; 3, 6, 9
Day 2: 1, 5, 9; 2, 6, 7; 3, 4, 8
Day 3: 1, 6, 8; 2, 4, 9; 3, 5, 7

It is possible to fill in the remaining days to get every pair together at least once. Necessarily some pairs will be together more than once over the 6 days. Here is something that seems pretty good:

Day 4: 1, 2, 4; 5, 6, 7; 3, 8, 9
Day 5: 1, 3, 5; 4, 6, 8; 2, 7, 9
Day 6: 2, 3, 6; 4, 5, 9; 1, 7, 8

Maybe something like this will work for you?
--Victoria