My favorite reading cloth is a cotton reading quilt.
They stay put, and they are easy to care for: washing machine and dryer.
Plus, every cotton reading quilt develops its own character and texture over time -- just like the cards.
I like to put a pattern on one side of the quilt and a solid color on the other side. Then, I lay the cards on the solid side, so that the pattern doesn't compete with them.
I've only recently learned how to apply a binding to a quilt. Previously, the raw edges have been fending for themselves, which has actually worked out quite well. There's a picture of one below, which I've been using for about a year. It has been machine washed and dried about a dozen times.