You can do this with a Wetjet style swiffer or even a regular sponge mop, too, you just need some more hardware. Clothespins will work, but these pape...
I had seen a few Swiffer instructables here, and thought what a great idea they were - buying the solution bottles and pads was expensive and wasteful, and the wet pads left a residue on the floor. I got all the stuff I needed, but didn't have time to make the washable pads or mod the wetjet bottle to make it refillable, so I came up with a quick and lazy way to get the same results.
All you need is a Swiffer, a bucket, some old washcloths, and your cleaning solution.
Step 1: Soap up the rag.
Soak the washcloth and squeeze it out, but not completely. The wetter it is, the more it will clean, but you don't want puddles. Lay the washcloth on the floor and center the head of the swiffer on it.