Andrew Krug has very kindly sent me a link to the Radio Free Europe North Caucasus languages service, which has archived RealAudio broadcasts in Chechen, Circassian, and Avar (not to mention Russian). I’m listening to the Circassian now; the only thing I recognize is “Adyghe Respublika” (adyghe being the self-designation for the languages called by foreigners Adyge and Kabardian/Circassian), but it’s fascinating to hear—it doesn’t sound anything like I had imagined.
Kabardian is famous for allegedly having only one vowel, but as this site says:
One striking feature of NWC [North-West Caucasian] is that the languages are very rich in consonants and very poor in vowels. In fact it is sometimes said that Kabardian has “only one vowel”. This is not actually true, but the peculiar phonology of the NWC languages makes it to some degree justifiable.