The Master and Margarita by Mikhail Bulgakov is my absolute favourite one! Its very different form Anna Karenina though, mixing philosophy with phantasmagoria and sarcasm.

Another one of my favourite authors is
Ivan Bunin. He mostly wrote short stories, rather than novels, but it is very very Russian in spirit. He's got the Nobel Prize in Literature.
Other popular choices would be
Doctor Zhivago by Boris Pasternak. Probably more in line with Anna Karenina. Pasternak was also awarded the Nobel prize, but was forced to refuse it by the Soviet government.