When I went to visit my then girlfriend's family in Russia a few years ago there was what was known as the $100 rule - basically that nobody ever left the house without $100 US in their pocket, because that amount would be enough to get you out of pretty much any trouble.

Bribery of police is so endemic in Russia that the state's most effective counter to this so far has been to simply bribe the police more and reclaim the losses from the perp - so for a traffic offence, the policeman would get a bonus from the state for not taking the bribe and the offender would have his car sold.

Paying police more in the first place might be a good start.