There really is no good schedule for 12 players in groups of 4 when there are only a handful of rounds, things get much easier if you can have 16 or more players. I have prepared the following schedule with your comment about wishing to balance each player's opponents as best possible, but this comes at the expense of having repeated partner pairs.
(A4 A3 v B1 B4) (A6 A5 v B2 B6) (A2 A1 v B3 B5)
(A3 A4 v B5 B2) (A5 A6 v B4 B3) (A2 A1 v B1 B6)
(A5 A4 v B3 B6) (A3 A1 v B2 B4) (A6 A2 v B1 B5)
(A1 A5 v B1 B5) (A4 A2 v B4 B2) (A6 A3 v B6 B3)
Team A has players A1 to A6, and team B has players B1 to B6. Each player opposes 4 members of the other team once, and the other 2 members twice. Hope that helps.