If they're kind of dirty and I'm just plopping them in a soup, I'll place them under running water, especially if they're just button mushrooms, or other tightly-closed fresh varietals.
If the gills are exposed, I'll gently pat/wipe them with a clean paper towel, regardless of the cooking method.
If I'm going to saute them and have to wash them thoroughly, I'll then put them in a salad spinner with a dry paper towel, then dry them off with a clean paper towel, keeping the gills as dry as possible beforehand.
If they are not going directly into a liquid, I'll then let them sit out near the warm stove if I can prepare them ahead of time, and/or time permits.
I'm sort of more concerned with the potential fingers that may have originally picked or handled them, in a plant that is less hygienically oriented than stricter ones, or the potential floors they've dropped on before I get them home, then I am with the dirt that may be on them, although depending on the amount of dirt, that's kind of a factor as well, as it may alter the flavor of some dishes.
I worked in the grocery business for many years, many moons ago, and saw some things I wish I hadn't by today's standards.
