Here's one way of doing it, where the 6 rounds are in the rows, and where after 4 rounds each player will have played with all others exactly once.
(1 2 3) (4 5 6) (7 8 9)
(1 4 7) (2 5 8) (3 6 9)
(1 5 9) (7 2 6) (4 8 3)
(1 8 6) (4 2 9) (7 5 3)
(9 4 5) (6 3 7) (2 1 8)
(9 6 2) (4 3 1) (5 7 8)