I always hear how they were fantastic, and they preserved the earth while we destroyed it, we gave then a bunch of diseases and waged war while they wanted peace. All things that we modern Americans disapprove of and put the Cherokee in a good light. But is there anything they did that would alarm us or we would dislike?
This could apply to other tribes, I just picked the Cherokee as I have truly beer heard something bad about them.
I don't know if this really counts, since you specify specifically the Cherokee, but one thing many people don't realize is how drastically indigenous peoples altered their local ecology.
There is debate right now (and it is a topic that has been considered even since Darwin's time) among anthropologists and archaeologists as to "what happened to the megafauna". Like, where are all of America's elephants, dire wolves, camels, wild oxen and lions? Most researchers agree that native American people killed them all or otherwise weakened their populations to such a degree that other natural causes drove these species' demise. The reason debate continues is that there is no concrete evidence in the archaeological record of this happenening -- i.e. there have been no megafauna bones found with a human-made spear sticking in it and, I suppose, no rock carvings unearthed with "hey guys, let's go kill all the megafauna" written on it.
In addition to these extinctions some researchers insist that even the formation of the vast, treeless prairies of central-North America have causes related to slash and burn practices of native peoples.
I mention these things because not only are many people today surprised that humans altered the Earth's ecology this way (similar extinctions are seen wherever hunter/gatherers migrated throughout pre-history) but it certainly provides a stark contrast to the romantic idea popular culture has of native american peoples living in harmony with their surroundings and nature, etc.
Again, this doesn't address the Cherokee people specifically, but certainly in a general sense indigenous American people contributed to the extinction of many animal species that used to be present here, just as indiginous people did in Asia, Europe, Africa and South America.
If you want more information about this I suggest you ask anthropologists (http://www.reddit.com/r/AskAnthropology) for the latest information regarding the megafaunal extinctions in North America and humans' role in environmental change in pre-history.
We hope that modern Americans would disapprove of slavery. The Cherokee were slave owners, owning about 4,000 black slaves in Indian Territory (Oklahoma) in 1860.
All of the "5 Civilized Tribes" in Indian Territory were slave owners, owning together about 10,000 slaves.