To supplement Andy's answer, there is a recent survey by Laura Demarco, "The conformal geometry of billiards," Bulletin AMS 48(1), Jan 2011, pp. 33-52. She defines a billiard table as ergodically optimal if, for each direction $\theta$, either every trajectory that avoids vertices is periodic, or every trajectory that avoids vertices is "uniformly distributed." It may be that your 'everywhere accessible' criterion is adequately captured by her definition of uniformly distributed. Ergodically optimal dynamics are also called Veech dichotomy.

Any billiard table that can be tiled by squares is ergodically optimal; in particular, the square is (every rational $\theta$ is periodic, every irrational $\theta$ will lead to the particle "spending equal time in regions with equal area" [modulo avoiding vertices]). The regular $n$-gon is ergodically optimal.

There are examples that have billiard trajectories that are dense but not uniformly distributed.

To supplement Andy's answer, there is a recent survey by Laura Demarco, "The conformal geometry of billiards," Bulletin AMS 48(1), Jan 2011, pp. 33-52. She defines a billiard table as ergodically optimal if, for each direction $\theta$, either every trajectory that avoids vertices is periodic, or every trajectory that avoids vertices is "uniformly distributed." It may be that your 'everywhere accessible' criterion is adequately captured by her definition of uniformly distributed. Ergodically optimal dynamics are also called Veech's dichotomy.

Any billiard table that can be tiled by squares is ergodically optimal; in particular, the square is (every rational $\theta$ is periodic, every irrational $\theta$ will lead to the particle "spending equal time in regions with equal area" [modulo avoiding vertices]). The regular $n$-gon is ergodically optimal.

There are examples that have billiard trajectories that are dense but not uniformly distributed.