I don't believe it is possible to have a schedule where all pairs play together in threesomes either two or three times each.
The next best thing is to have pairings either two or four times each. For example:
(1 4 2) (3 6 5)
(4 1 3) (2 5 6)
(1 5 4) (6 3 2)
(4 6 1) (5 2 3)
(2 1 5) (3 4 6)
(5 2 4) (6 3 1)
where the three pairs (1 4) (2 5) (3 6) are the ones that occur four times each.
