I've never had any luck keeping my feet dry in a real downpour, but neoprene covers do pretty well at keeping them warm. Anything windproof is a big help in the warmth department, so most covers will help.
Your choice of shoes and socks can go a long way toward keeping you comfortable when wet. Some combinations feel wetter and take longer to dry than others.
Something else should be mentioned... Fenders will help keep your feet much less wet since often the bulk of the water comes from the front tire.