If you're talking about a real devilish downpour, I doubt if it is possible to stay dry. you will probably have to change after youreach your destination. Even then, the "dry" clothes you have in your backpack or pannier might be damp too, even if you had the foresight to put them in a plastic bag. At least cycling clothes and other exercize clothes dry pretty fast.

My biggest problem--and it can be quite dangerous--is that my glasses get so wet that I can't see. I don't know what to do about that!