I don't expect Kino to get mentioned for a long, long time if this thread lasts that long, mostly because it's not popular among the non-Russian crowd. But within the Russian crowd, jeez, they're better than the Beatles, they actually caused major damage to the USSR with their message. More than 20 years later and you still hear their songs all over the damned place, at least in the Russian-American community I live in. There's a reason for that, too. So they're getting my vote, and not because of their political consequence. Their music is the truest, maybe not the most talented, but the one that connects with the listener the most.
I'm not going to vote for them, but I'm also going to mention DDT as a Russian rock group. To give you an idea of the popularity there, their lead, Yuri Shevchuk, started an argument with Vladimir fucking Putin at the dinner table and got away with it. They're also one of those groups that may not be the most talented, though they're certainly very talented regardless, but also connect with the listener. They use culture, folk influence, language play. I just don't see that from most of the rock and metal groups mentioned so far, but I suppose most people don't care about having petty things like "cultural influence" in our music, everyone wants to bang that shit out.
As for a band most people actually understand, mainly because there isn't a fucking language barrier in the way, I'm going to go with Motorhead, also because that hasn't been mentioned much, and they need their due.