As Jim said, unless it's a gully washer or there is lightning, high wind or other things that would make it dangerous, fishing in the rain can be quite enjoyable.
I only get to the mountains a few times a year but around here in Kentucky, I fish a lot when it rains and have found fish frequently feed quite heavily on the bounty the rain has washed into the streams and lakes.
As I told my son this year when we were fishing and a rain came up. The fish are already wet, they don't mind the rain.
Now, when it's really windy, I don't like being out risking a tree falling on my head, and I'm no fan of waving a 9' long lightning rod around over my head when the sky starts flashing. Also, when I'm stream fishing, I keep an eye on what the sky looks like upstream of where I'm fishing.
Flash floods are a real danger when fishing running water. Keep an eye out for rising water, unusual amounts of debris floating in the water, a sudden subtle change in water color, anything that looks or sounds abnormal. If you see it, get out and make sure you're getting out on the right side of the stream.
If it was nothing, in a while you'll know and can always start fishing again, but if it is a flash flood coming, you don't want to get stranded on the wrong side of the stream. It could be miles before you find a place to cross.