Depends on the temp.Tights will be too hot for most people except in the depths of winter. If it's cold I'll wear hat and gloves, some people can get away with not. I can't.
3/4 leggings or shorts of the lycra varity. They dry quick - well they don't really absorb the rain in the first place so they are dry quicker. I have wind resistant / water resistant tops which are good for most days. One is sposed to be for summer but it's far too warm to wear except in winter.
You'll not often find yourself in a torrential downpoor and if it is that bad you can postpone your run for an hr when it should have eased off.
Doesn't really matter too much in the summer as you'll not really get cold. Long runs in the wet in winter are a pain. Summer runs aren't too bad.