Hmmm...good question.
To be fair, I'm only part Native American (Cherokee)...and I still suspect I'm part African, but that's another story.
My former college used "Indian" names (Indians for the men, Otahkis for the women) for the sports teams. And of course, you have the Braves, Blackhawks, etc...and the Fighting Illini of the University of Illinois.
The Sioux were some mean-spirited fightin' mofos from what I've read about them. But I don't have a real problem with the "Indian" names, with the exception of the Washington Redskins. (I mean, come on...that's just not cool anymore. You might as well have a team like the Philadelphia Darkies or something.)
The problem I have with the Native names involves things like the tomahawk chop in Atlanta and Chief Wahoo in Cleveland. That sorta thing used to not bother me, but as I read up more on Native American culture and what not, I can see why those type of things are considered offensive. Now you won't see me raise all hell over it, but it does make me cringe.