The first question after Kosovo's independence is what will happen to the ethnic Serbian minority. After the war there were reprisal attacks and most of the Serbs who stayed moved to the northern part of the country to be close to Serbia proper. These Serbs refuse to participate with the ethnic Albanian government and want to remain in Serbia. Serbia and Russia will probably push for either an autonomous zone or secession to Serbia for the Serbian zone.
Spreading of violence is another concern. Already Albanian terrorist groups have waged war against Macedonia and sought to annex part of Serbia proper. On the flip side there is rising fear of Russians using shadowy means to do hit-and-run attacks for the Serbs against the Albanians.
Geopolitically there is a question of precedence. If it is okay for Kosovo to unilaterally declare independence then what is stopping the Serbian political entity in Bosnia, the Republika Srpska. Russia may then push South Ossetia and Abkhazia into formalizing their independence and into Russia's geopolitical orbit. Elsewhere there are also concerns. While the drive for Kurdish independence from Iraq has tapered off as of late there will always be the Kosovo model to look at.