What the heck, we are on a roll here, I thought I would try this.
Here is my preliminary list:
1961 Yankees (see Bill James, open and shut case as he lays it out)
1970-72 Big Red Machine (not the later edition)
1986 Mets (Possibly greatest talent of any team ever. Possibly biggest waste of talent of any team ever. And yet, we loved them at the time - at least in NY).
1972-4 Oakland Athletics - Their stats don't seem amazing, in part because the era had a neat balance of pitching-hitting, no steroids (that we know of) and pitchers' parks. Also, there were other good teams. But the As were better and played differently than everyone else. They were as much fun to watch as any team I can recall.
1969 Mets (won 100 games, overcame great Cubs team to beat them by 8 games in division, swept Atlanta team with Niekro, Wilhelm, Aaron, Cepeda, won 4 of 5 from extremely powerful Orioles team. It was not just a miracle).
1999 Mets - better by far than the 2000 team, got psyched out early in that series with Atlanta, then nearly pulled of a 2004...One of the best fielding teams, especially IF ever (not counting Piazza).
2000-2001 Oakland Athletics. Should by all rights have succeeded the Yankees Dynasty. Best team in baseball both those years. Like '99 Mets v. Braves, they got psyched out by Yankees' mystique.
1976 Yankees - better than the next two seasons when they won the Series. They were tired and emotionally drained coming into the Series v. the Reds, and probably couldn't have won under any circumstances, but would have put up more of a fight. Had to start Doyle Alexander in game 1 for example. Not until 1996 did a Yankees' team play with such verve and joie de vive.
1967 Cardinals. Not that many teams whose starting lineups and pitching rotations I can remember entire years later. They are one. Individually they don't seem that awesome, but they were very good. Gibson made them great.