I'm sorry I'm 3 years late, but I have a set of three dice with slightly better overall probability than the ones you have. The dice (1,1,4,4,4,4), (3,3,3,3,3,3), (2,2,2,2,5,5) (I'll call them A, B, and C respectively) have an overall probability of 17/27=62.96%, slightly better than your 67/108 =62.04%. A beats B with probability 2/3, B beats C with probability 2/3, and C beats A with probability 5/9.
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