The blue painter's tape is what I used to tape it to the floor, and for many other things - I love the stuff. I was thinking that it wouldn't work as well on the table because the quilt would be wider than the table and I'd have to tape it to the underside, but the bottom of the table is not flat, it has a 2 1/2" vertical edge all around it, about 1 1/2" in (why is it so hard to describe something relatively simple?).
Maybe tape for smaller quilts and just letting in hang, as Sewnoma suggested for larger ones. I know I'll have to try it several different ways before I figure out which one works best for me, and I really appreciate all the suggestions! Deb Watkins, the rubber grip clips sound like a good idea, and I have a BBB coupon, making it even better.