Paper towels here, too, because I don't like putting anything that is oily or greasy into my washing machine.
So the oily residue gets a good wiping with a paper towel. I re-use paper towels until they can't be re-used again. Sometimes 6-10 times, depending on the lighter-use events. If they get wet with only water or tea, I let them dry before the next use unless I need to clean up a messy or very dirty spill. The final usage tends to be to wipe out oily, greasy residue inside a frying pan, the wok, or a soaping vessel. Since going to a much lower SF, though, my soaping vessels don't end up being as oily as they used to be in the beginning.
My niece, an avid crafter, mostly with needlework related things, gave me a roll of re-usable cloth (cotton flannel) paper-towel substitute. They have plastic snaps on each end and each 'towel' snaps to the next one. The purpose is to put them on a paper towel roll and thus never have to toss out a paper towel again. Difficult as they are to use (snaps just don't disengage with a flip of the wrist they way a scored paper towel does from the roll), and thier limited absorbancy (lightweight flannel), I still like the idea of them. And I do use them in the soaping area, mostly to spread out to protect surfaces from batter splatter, but also to clean-up as needed, while avoiding saturating them with oily residue (because then I'd have to toss them in the trash.)