There was an upper class accent. While the communists did not intend to wipe it out (Lenin is speaking it in all of his records) it fell out of use except in movies portraying emigrants and the surviving intelligentsia. The distinguishing characteristic is a French or German "r" sound rather than the rolled Russian R.
Most Soviet leaders sought to underline their humble background by speaking with slight dialect influence from whereever they came from. Stalin spoke as a Georgian guest worker. Brezhnev and Khrushchev pronounced the G as a H, like they do in the South and in Ukraine, but none of it had an influence on the Russian language as spoken by Russians in general.
Their speech was mostly tainted by the Muscowite dialect. The Communists moved the capital to Moscow from St.Pete's and features previously identified (by Lomonosov) as the Muscowite dialect (unstressed o and unstressed e are pronounced as ah and ee) are now standard Russian.
There were two dialects that might be thought of as "posh" in some sense. Even while the capital was in St Petersburg, Moscow set the standard. Old Moscow Pronunciation (OM) had several distinguishing characteristics, which share with each other that they are divergences from Russian orthography. Notably, ходят 'they go' was pronounced with [u] in the second syllable, not the [i] we might expect on the basis of orthography and processes of vowel reduction. There was also a common habit of pronouncing the masculine singular adjective ending -ий as [əj], not the expected [ij]. We can see this pronunciation reflected in the first line of Lermontov's [well-known poem Парус](http://ru.wikisource.org/wiki/%D0%9F%D0%B0%D1%80%D1%83%D1%81_(%D0%9B%D0%B5%D1%80%D0%BC%D0%BE%D0%BD%D1%82%D0%BE%D0%B2)), where what is nowadays spelled одинокий and pronounced roughly [ɐdʲi'nokij] was spelled одинокой. Another feature of OM that is nowadays unremarkable is the pronunciation of the letter щ and letter combination сч as /ʃʲʃʲ/. In OPb, these would have been /ʃʲtʃʲ/ both. OM also turned the stops /k/ and /g/ into fricatives /x/ and /ɣ/ before other stops, so that words like /kto/ turned into /xto/. Most of these pronunciations are no longer common at all, though a few have entrenched themselves into Contemporary Standard Russian. There are a lot of other changes that have happened in Standard Russian since the fall of OM pronunciation, but I refer the curious reader to the first chapter of Comrie, Stone, and Polinsky The Russian Language in the 20th Century.