If I have understood you correctly, then I can't see a solution to the problem as stated, however I can make it work if you have 8 rinks and two of them vacant in each round.
round rink1 rink2 rink3 rink4 rink5 rink6 rink7 rink8
1 (A1 B1) (C1 D1) (A2 B2) (C2 D2) (A3 B3) (C3 D3) ( ) ( )
2 (E1 F1) (G1 H1) (E2 F2) (G2 H2) (E3 F3) (G3 H3) ( ) ( )
3 ( ) ( ) (A1 G1) (B1 H1) (A2 G2) (B2 H2) (A3 G3) (B3 H3)
4 ( ) ( ) (C1 E1) (D1 F1) (C2 E2) (D2 F2) (C3 E3) (D3 F3)
5 (C1 H1) (A1 F1) (C2 H2) (A2 F2) (C3 H3) (A3 F3) ( ) ( )
6 (D1 G1) (B1 E1) (D2 G2) (B2 E2) (D3 G3) (B3 E3) ( ) ( )
7 ( ) ( ) (B1 F1) (A1 E1) (B2 F2) (A2 E2) (B3 F3) (A3 E3)
8 ( ) ( ) (D1 H1) (C1 G1) (D2 H2) (C2 G2) (D3 H3) (C3 G3)
9 (A3 D3) (F3 G3) ( ) ( ) (A1 D1) (F1 G1) (A2 D2) (F2 G2)
10 (B3 C3) (E3 H3) ( ) ( ) (B1 C1) (E1 H1) (B2 C2) (E2 H2)
11 (A2 H2) (D2 E2) (A3 H3) (D3 E3) ( ) ( ) (A1 H1) (D1 E1)
12 (B2 G2) (C2 F2) (B3 G3) (C3 F3) ( ) ( ) (B1 G1) (C1 F1)
13 (E3 G3) (A3 C3) ( ) ( ) (E1 G1) (A1 C1) (E2 G2) (A2 C2)
14 (F3 H3) (B3 D3) ( ) ( ) (F1 H1) (B1 D1) (F2 H2) (B2 D2)
Does that help? If you only have 7 rinks then I guess you could assign the rink8 matches to one of the vacant ones, you will end up with some people playing twice on a rink, but it will not be too badly unbalanced.
