Tolstoy was very well recieved by the Soviets, despite the fact that he was a believer and an anarchist -- they liked very much the fact that he hated the clergy (he was excommunicated from the Russian Orthodox Church) and private property. In fact, the 1967 production of War and Peace was the most expensive film ever made in the Soviet Union.

Dostoyevsky, OTOH, was banned under the Soviets until the 1970s; read, say, The Idiot and it's not hard to tell why.

Chekhov was very well respected in the Soviet Union mostly because a large amount of his work is to do with the collapse of the aristocracy of Russia. His plays were performed throughout the era with major productions in the Moscow Art Theatre in the 1930s.

It's not Chekov. It's Chekhov. Or: Tshehhov. I will follow Nabokov's Onegin translation in transliterating the cyrillic x as a Latin x. It's the "ch" in "Loch Ness" and not (perish the thought!) a "k".

Stalin was more conservative than his Bolshevik predecessors. He did reintroduce tsar era uniforms and punishment for homosexuality but Dostoyevsky didn't have to be reintroduced. He was never removed from the curriculum; Instead the communist had changed the approach. In the 1928 literature schoolbook by M. A. Rybnikova for the Brothers Karamazov you are, for instance, invited to write about "how the social question is portrayed in the character of Snegiriov"; another task invites the pupil to attempt to do a class analysis for all the major characters.

The "Socialist Realist" -Ostrovskiy, Gorkiy, Fadeyev, Sholoxov- writers were sort of the official art movement of the Stalin era since they were, along with pulp, the only kind of contemporary fiction not banned after the yezhovshchina terror^1 . They mainly tried writing like a Thomas Mann, Theodore Dreiser, Dostoyevsky, Balzac and Tolstoy, but with a socialist slant. The modernists, i.e. folks who didn't like Tolstoy, Chexov & Dostoyevsky, instead got murdered or went to the camps. Examples: Babel, Charms, Erdman, Meyerhold, Pilniak, Zabolotskiy & Eyzenshteyn.

So that you can count to paying tribute to the classics, too.

^1 on Soviet censorship there are these two nice articles albeit in Russian:

The Archive Historian Mixail Vladimirovitsh Zelenov Glavlit wrote additionally a very detailed work on the censorship conducted by the Glavlit. Before 1932: local authorities go berzerk on books they personally do no like. Volin and Krupskaya stop it. Before 1936: censorship is centralised, rare and mostly concerns non-fiction by the now-defunct ex-comrades of Stalin. 1936-40 25 million books get destroyed across the USSR.

the Russian censorship historian Arlen Blyum (Арлен Блюм) on the sheer extent of the last bit: the mention or depiction (f.e. on a group photography) of repressed people or their relatives did warrant destruction