In the past, my wife and I have been using a piece of tarp, but it's definitely not ideal. It's loud and provides no insulation from the ground.
My wife likes to quilt and we came up with the idea of quilting a 3'x3' or so pad using coated silnylon on the bottom and cotton on top. It would be a little heavier, but would be much more comfortable for the dogs. Protects some against the cold ground, protects against moisture from the ground and would be washable.
I've seen Fronkey's video where he uses a piece of building wrap, I believe it was. Is that what most use or have any of you come up with some clever ideas?