"The Serbian population [of Kosovo] strongly object to the illegitimate declaration of independence. Their protest actions are on the rise, in particular in crowded areas of the Serbian province, and in southern [Serb] enclaves," said spokesman Mikhail Kamynin.
The situation is of "serious concern" to Russia, he added.
Kosovo has been a UN protectorate since NATO bombings ended a conflict between Serb and Albanian forces in 1999.
Kosovo's independence has been recognized by the United States, Australia, Japan and major European countries, but Serbia and Russia have reacted angrily, with Serbia, which sees Kosovo as its historical heartland, recalling its ambassadors from those countries.
Moscow argues that Kosovo's independence will trigger more secessions and that it will block it from joining the UN.