I'll admit, I didn't watch TNA during the time period America's Most Wanted were active, so my input is going to be a little one-sided. I have, however seen the Steel Cage match everyone talks about, as well as all the matches leading up to that, and some that followed.
That being said, with my limited knowledge of AMW, I'm making my decision based off what I perceive to be each teams' greatest rivalry. AMW had Triple X, and one can make a case for a few others. That's the one I know, so that's the one they get. Sorry...about your damn luck.
If that's the case, then I have to give it to Beer Money for their Best-of-Five series with the Motor City Machine Guns. Even though they ended up losing that the final match, the simple fact is that they put on some of the best tag matches I've ever seen, in some of the best match types possible. One feud yielded us a Steel Cage, the Ultimate X, a ladder match, a street fight, and a 2-out-of-3 falls? It's a shame TNA didn't care abotu that more once it got started. It's ridiculous that it ended on a free episode of Impact, and that they cared more about the EV2 tie-in than great wrestling...
Like I said, I really only know Beer Money. But while they were "on", they were REALLY "on".