Which white covers do you speak of? Army? Navy? Dixie Cup? Combo cover?
I have found that for the Navy Combo cover the easiest way to clean it when it gets dirty is to just buy a new one. That part of the cover isn't that much and is just easier once it gets grimy.
Also it comes in two flavors, one is more cloth and the other is more almost vinyl. For the cloth one I've used those small pen-like stain removers and similar products with "okay" success.