I am with you, I just throw wet stuff in with the rest of the laundry. On the off chance I don't think I will have any laundry to do for a few days I hang up the wet stuff on my indoor drying rack to keep them from mildewing. In my house a bag, even a mesh bag, full of wet clothes, napkins, whatever would start to smell or mildew within 24-36 hours. Heck if I leave my laundry in the washer for too long it starts to smell.
Where I live, things do mildew pretty fast. That's why I am willing to wash wet things with any other types of clothes. I throw them in the bottom of the washer and just wash with the next load.
I was M, W, and F every week, with the last load being in the late afternoon (with diapers and all, I usually do about 3 loads a day on those days). During the weekend, I will do a load when the washer fills up (and it usually does, between napkins and dirty gardening clothes and maybe a wet shirt from milk being spilled at supper and whatever else gets dirty and wet). So, at most, my wet napkins are going from dinner on, say, M, to W am, which is about 36 hours. In that time, I usually don't have a problem with mildew.
If I waited a week to have a load just dedicated to kitchen laundry, then, yes, my stuff would mildew. I live in a hot, humid climate, though, which makes a difference. We battle mildew and mold even in the dead of winter with forced air heat.