In most cases the biggest issue with rain is psychological—there’s just something about heading out into a downpour that’s so much worse than if the rain were to start falling once you’re out. I always try to remind myself that you can get only so wet, and you’ll reach that state of saturation within the first 10 minutes.
If it’s a cold rain, there’s no getting around it—the run is unlikely to be one of the highlights of your life. That’s one of the few times I dress for how it’s going to feel stepping out the door, because even if you do warm up, your clothes will be drenched by then, so a little bit of extra weight won’t really matter.
I always make sure my shoes are tied snugly when it’s raining. If they’re too loose, the rain is going to cause your feet to slip around some, and your knees or other parts of your leg can get achy before the run is over from the extra motion.