How about a block keeper? I have a 14 inch square of felt like material that a 14 inch square of thicker gauge tablecloth plastic is sewn onto. The plastic is layed on top of the pinked felt square and a 1/4 inch seam is sewn on one side. A ribbon is sewn on the felt on the opposite side, in the middle. You put your unsewn block pieces in it to travel with. I've used mine a lot for going to classes for both sewn and unsewn blocks. You roll it up and tie the ribbon in a bow.
I have also used it for a quick quilting design trial template. I put my finished block under the plastic and use a dry erase marker to try out quilt designs.If you used different coloured felt and rolled them up, they would look cute in a container in the middle of the table.