In case you haven't noticed, the whole country has gone way overboard on political correctness. Our high school teams did not get the message which makes me sad. But, I'm here to help:
Eagles, Falcons, or any other bird of prey: Raptors
Indians or indigenous peoples: First Nations
Bronc(h)os, Stallions, or anything horse-related--Equines
Bison/ Buffalo: Bovines
Bulldogs: Domesticated Canines
Wolves: Largest Canines
Cavaliers, Patriots: Defenders, Protectors
Ancient (Athenian, Spartan, Trojan, etc.): Big Shot, Leviathan
And the two nearest and dearest to my heart:
Rebels: Agitators, Insurgents, Renegades
Devils: Kings of Hell, Lucifer, Holly Terror, Scallywag, Wretch
Your team may not be here, but don't think they're totally innocent, I just haven't come up with a name for them yet such as Greyhounds: sporting canine and Giants: Mutants
Let's have them...