I chase cold fronts. It often leads to hunting in rain. I won't hunt in a straight up downpour, unless some unique circumstances would arise. But my usual encounter with rain is heading out when everyone else is looking at the cloud on their iphone weather app and assuming its just "raining". I try to be in the woods for as much of the 48 hours following a front passing through as possible. This often means walking to a set, scouting, stalking, or setting up in rain, as it is beginning to let up. Sometimes I'm off on my timing a little or have a long trek, and I'm heading in while it's pouring.
I take shorter shots. I don't climb difficult trees, if I climb at all. I minimize any risk of injury, or of not putting a broadhead where it needs to be.
But taking these things into consideration ahead of time, and knowing that in the southeast (everywhere, really) deer will move more in daylight with big swings in temperature (cold fronts), I don't even think about rain. It is not part of the equation, unless as I said before, it will just be pouring for hours.