What about the eastern and far eastern portions? Like Siberia and further east? I don't know how well held together that region and east is as far as connection to Russian culture.
Cechenya and Dagestan will definetly sieze the moment and go independent.
It would be just the beginning of the Balkanization of Russia and losing large portions of territory.
Oh yeah, I just realized while typing, there is the issue of any nukes in the breakaway territories.
There are many Republics further East that should generally belong to their respective ethnic minorities e.g. Buryatia, the Altai Republic and so on, but there are similar numbers of Russians in those Republics as there are in Crimea now so I would imagine they would stay close to Russia. However they would probably seek greater autonomy in general over there because seemingly a lot of Far Eastern Russians are unhappy with the over-centralisation in the country.
To put it in perspective, I remember when I asked a Russian friend of mine where he was from he said "Central Russia" so I said "Wow, do you mean like around Novosibirsk?" and he said "No, near Moscow".
Tuva is one of the only places outside of the Caucasus that I think could conceivably secede. They are basically independent right now in that they have a tiny Russian minority there (comparatively), they're predominantly Buddhist, they used to have a lot of trappings of an independent state back in the '90s (they were de facto independent like Tatarstan) and they're so far from Russia and so close to Mongolia there's really no reason to be a part of Russia.
Maybe the US can sign a non-proliferation treaty with them under terms that we promise to never, ever be hostile in return for their giving up nukes?
I'm sure everyone will work it out. Kazakhstan briefly held a hilarious quantity of nuclear weapons when they became independent. It's a miracle Islamic extremists were so stupid and single minded in their attempts to rule Afghanistan/take over Tajikistan and the like that they never tried to take any of it.