I composed the Native American city list and as many have pointed out, I went for genuine places that usually represent real, important, sizable and durable settlements, the closest things the Native Americans had to real cities (but at the same time since the Native Americans are supposed to represent all Native American tribes I tried to ensure there was sufficient variety as well: while most names are either mound builder or pueblo sites (as those were the most important ones historically), settlements from tribes as varying as the Canadian Inuit, the Californian Ohlone, Florida's Apalachee and Maine's Abenaki are present).
This often forced me to use European names as the native names of these places are not known to us today. That's unfortunate but there's no perfect solution for this civ anyway, this whole civ is problematic to begin with (although IMO still better than the Sioux who never had ANY cities or anything close to it -- they weren't a civilization by any stretch of the definition, but that's been discussed to death already). IMO it's not much worse than some of the other civs: the Egyptians and Persians have mostly Greek and Arabic names, the Carthaginians and Celts Roman ones, the Aztec and Incan lists have plenty of Spanish names on their lists, etc. Not to mention that many modern civs like the Germans and Russians have their names anglified (Cologne instead of Köln, Moscow instead of Moskva, etc). It of course stands out more with the Native Americans but a lot of civs have names that aren't authentic.
You could argue that native names are more authentic but problem with that is that there are virtually NO sizable and durable Native American settlements if which we know their native name (even many of the 'native' names on my list are the names that modern NA tribes or even modern American historians gave to these sites, not their original names). You'd have to resort to making sh*t up or just using random terms that have no historic significance of any kind, as has been done so often by Firaxis in the past -- but I feel that's got to be the worst solution of all (I've worked hard to remove the leftovers of that from Civ4: I recreated the Zulu and Indian city lists from scratch as well as all the lists in BtS and some of the other ones in Warlords and before). This is a history game, the city lists should be based on real history.
In the end it's a matter of personal preference though. I'm sure plenty of mods for the Native Americans will appear, some to just replace the city names with more authentic-sounding ones or tribe names, some to change the civ to the Sioux or Iroquois and some will simple remove it from the game entirely. I think those are all valid ways of dealing with this issue, everyone will have to decide for themselves what they like best.