I've been using cotton cloth diapers on top of some cheap white dish towels (NOT terry cloth) that I got at Walmart just for this purpose. I got the "flour sack" towels, which aren't super absorbent, but they are 100% cotton and a very fine weave, so no strings. I got another kind that was a cotton blend, also a fine weave, but a little more absorbent than the flour sack towels. They are working great. I forgot how messy ducklings are.
That was actually my concern about a dish towel... strings and fuzzy tips. But I agree about traction being so critical and cheese cloth (and paper towels for that matter) do not provide that. QJ, I will look for rubber shelf liner. If I have that, I may try that instead of the towel.