I've spent the last several months (or is it 3 years?) listening to Chechen language and music, and reading their history. It's possible a connection exists between Caucasians (i.e., natives of the Caucusus) and other lost/isolated peoples around Eurasia. It's also possible these are the last unassimilated survivors of an ancient, but lost civilization (maybe Sumerians, someone suggested), and that the rest of the population mostly disappeared by blending in with neighbors.
Anyway, it's interesting to hear and think about. Check out www.chechnyafree.ru, and visit their music pages. May I suggest Aza and Marina Aidaeva.
As near as I can tell, the consensus is that the Northern Caucasian family (which is the Caucasian in Dene-Caucasian) is not closely related to Kartvelian (Georgian, etc aka Southern Caucasian), and that the similarities between them are contact phenomena, like exist in the Balkans or in Africa (where a few Bantu languages have adopted "click" sounds from Khoi-San)