Two soldiers were killed when their vehicle was attacked near the village of Alkhan-Yurt in southeastern Chechnya.
A third fatality was a member of the Vostok battalion, which comprises Chechens under Russian command. He died in clashes with rebels in the southeast. A Chechen guerrilla was also killed in the clashes.
An unidentified official with the pro-Russian Chechen administration said three other soldiers were killed in a series of attacks by Chechen separatists.
Despite Russian statements that the conflict in Chechnya is over, there are almost daily firefights between some 80,000 troops and a mixture of Chechen independence fighters and Islamic militants.