Lenin didn't have an accent but he had dyslalia and he couldn't correctly pronounce russian "r" (russian "r" is much stronger than english one). This defect obviously was a source for some jokes but it's not very unusual for Russians, everybody knows at least someone who can't pronounce russian "r" (p).

You can hear Lenin's real voice here

Stalin had a distinct georgian accent (actually, it sounds like armenian accent or accent from someone from Caucasus but I don't want to use word caucasian here). Every russian speaker could feel this accent but this accent is quite frequent and recognizable in Russia, so it wasn't a really big deal. This accent didn't sound alien for the russian ear and Stalin's speeches are completely understandable.

You can hear Stalin's real voice here

The third USSR ruler with quite an accent was Gorbachov. For the first years of his rule he had a distinct southern accent (this accent is popular in Russia's south and in Ukraine). The main difference from the 'norm' is using soft г instead of hard one. Very roughly it's like someone is saying 'hhain' instead of 'gain' but not quite.

Their manners of speech are easily recognizable by russian people. The same could be said about Putin (no accent, distinct manner of speech), Brezhnev (no accent, but not so good diction for his last years) and Eltsin (no accent, distinct voice and manner of speech). Actually, there's just one russian/Soviet ruler whose voice and/or manner of speech is not unique or widely known — Nikita Khrushchev.