I think the hardest part about dealing with this issue is finding out about it. Mostly, it is done in secret and both the giving and receiving end are okay with it. That makes it really hard for authorities to find out and try people for bribery and corruption because it will be hard to prove it.
I also don't understand how grease money can be seen as "normal" or "acceptable." If it was normal, we wouldn't even be discussing it. It's clearly out of the ordinary and unfair to people who can't afford it or don't want to give it.
I know of an NGO in India that is trying to fight against grease money there by getting people to give out paper monies instead of real money to get their government work done. The paper money has written on it "I refuse to bribe." It's actually a growing movement and I think it's great!