Well, it opens up a political and social can of worms, but the native population has had problems in adjusting to Western life. Also, the Danish socialist planners probably made things worse.
I have seen photos of Danish style suburban low-rise aparment buildings with seal skins hanging over the balcony. Alcoholism is rampant and I have heard crazy stories..... The murder rate is very high (on par with a US ghetto ca. 1986) but a visitor shouldn't feel threatened since it's probably internal alcohol fueled quarrels.
So, I would go for it if you are going spend time in the wilderness but it doesn't sound too pleasant if you are going stay in a town like Godthab (Nuuk), etc.
A quick Google search yielded stuff like:
http://nativenet.uthscsa.edu/archive/nl/9508/0081.html
p.s. When I was a kid my parents won a free trip to Greenland on SK. This was before the 1985 autonomy.... "Um, could we change that for something else?", "Well, we consider this to be a domestic Danish flight" (Great... a 20 minute flight to Aarhus) "Hmmm... but SAS is the national carrier of three countries..." "Ok, you can fly anywhere within Scandinavia"