Try this (Marna's trick):
Pour some bleach and water into a bucket. Pick up a hand full of hay or straw. (Of course it doesn't have to be clean. Just some nearby trash hay will do.)
Submerge hay/straw in mixture and use to scrub away grim. Hay works great at taking the grim off and gets down into all the nooks and crannies of a bucket. Much better than a brush I might add.
Follow up with some good rinsing and you're good to go!
Let me know how it goes!
