Here are my own solutions I put together. They have already been equaled or surpassed by Daniel's answer, so I'm just posting them for comparison. Leaving out rooks for now until someone else posts their own solution.
Bishops:
14 of each can fit:
. B . A . C . . . B A A B C C .
A . C . B . A . A B C C B A A B
. C . . . . . B . C C . . . A B
. . B . . . . . . . B A . C . .
. . . A . C . . . . B A . C . .
. . C . . . A . . C C . . . A B
. B . C . A . B A B C C B A A B
. . A . B . C . . B A A B C C .
Just one color Both together
Knights:
10 of each can fit:
A C B . . B C A
A C B . . B C A
B C A . . A C B
B C A . . A C B
. . . . . . . .
. . . A A . . .
. . C . . C . .
. . . B B . . .
Queens:
6 of each can fit:
A B . . B A . .
C . . . . C . .
C . . . . C . .
A B . . B A . .
. . . . . . . .
A . . . . . . .
. . . . B . . .
B . C C A . . .
Kings:
12 of each can fit:
A B B A . A B .
C . . C . C . B
C . . C . . C A
A B B A . . . .
. . . . A C C A
A B . . B . . B
C . B . B . . B
. C A . A C C A