Cleaning up icky messes without paper towels

We got a great email from Jennifer who has successfully kicked paper napkins and towels out of the house and has converted to cloth (much to the horror of her paper towel-addicted mother...her mom now brings her own paper towels when she visits!). Hooray Jennifer, and no doubt your dear mama will one day see the light as well! Recently Jennifer had her first icky clean-up mess to deal with and resisted the paper towel temptation (she still has some left). She writes:

For the first time since my newfound ways a family member got sick, several feet from the bathroom. Unfortunately, I had a pile of things that were waiting for me to take downstairs. And my trusty paper towel clean-up method was GONE! I had nowhere to turn except my kitchen towels. I washed them twice and still had to clean out my washing machine from bits of partially processed food. (Sorry for being so descriptive.)
 
I’m confident it’s probably a no-brainer…but I’m puzzled. What to do in this situation? Do you soak your clean-up towels in a bucket or something first?

This, I believe, is a common predicament and a big challenge. I would probably reach for my 'emergency' roll of paper towels (which I've had for over two years now), but I know there are less wasteful methods. Would you soak your towels in a bucket and empty it in the yard or the toilet? Soak them in the toilet? In your kitchen sink? Or do you give yourself a pass (and a break) and go for paper?