I've done my share of running in the rain, and I generally gravitate towards my Brooks Night Life hat. It's essentially a standard ball cap-style hat, but it's bright yellow and has a red LED flasher built into the back of the hat (both for visibility, of course). It does a nice job of keeping the rain off my head and out of my eyes and the visor helps me protect my vision from oncoming cars with bright headlights (many/most of my runs are in the dark). If offers no ear protection (from cold or sound, I suppose), however.

It sounds a little bit like you might be looking for a winter hat. For the most part, if it's raining during the winter, I've just added a pair of Ear-Grips or a fleece headband thing to keep my ears warm. Otherwise, you might want to take a look at some cycling caps. There are quite a few out there that have fold-down ear flaps, though the visors tend to be a little bit shorter. For example: http://www.nashbar.com/bikes/Product_10053_10052_501226_-1_201579_10000_200490

Don