I have problems remembering right and left! I am not joking. Somehow, I always think that 左 zuǒ is right.
To remember the Chinese characters, I will associate the radical kǒu 口 (mouth) with the right since almost every one uses the right hand to eat :D
I have no idea why the radical gōng 工 (work, labour) is associated with the left hand.
I wonder if it's got anything to do with the old Chinese superstitious belief that left-handed people are not good?
I know a few left-handed people who had been beaten and punished when they were kids learning how to write. The poor things were always beaten or scolded by their Chinese grandmothers, mothers and even fathers if they attempted to write with their left hands!
Whatever the reason, I MUST remember that 右 yòu is RIGHT (hehe, "You" are right!") because I have caused many a taxi driver to slam on the brakes when I tell him to turn into a dead end :S