I'm experiencing the exact same thing. It seems like it started happening in the past few weeks. I have an AppleTV 2 and an AppleTV 3. Both have awful picture quality, hang, and are virtually unwatchable. Resetting and restoring don't fix the problem. I am able to watch the same game, on the same network, in near perfect quality on iPad, iPhone or desktop computer just fine. Network bandwidth tests on the AppleTV don't indicate a problem and my wireless reception is "excellent." This is clearly an AppleTV problem from MLB.
To top it all off, MLB has started consistently showing commercials between innings. AppleTV seemed like the last outpost of serentiy where the silence between innings was a feature worth the cost of the service. Add obnoxious, repetitive commercials to the lack of quality from mlb.tv and it is all plain unacceptable.
MLB: fix the picture quality.
Apple: Make them dump the commericals.