I've watched a lot of Eleanor Burns quilt tutorials on her website and she uses Dragonfly's (Lynn's) method by holding the threads to the back as the fabric feeds through. I've also seen her and others start with a small piece of fabric that is sewn first (2x2 maybe). Don't cut the threads but immediately start feeding your fabric pieces through after it. After you've finished sewing you just cut off that starter fabric. You can use that same starter over and over. I would use two pieces of starter fabric layered together. I think these are two really good ways to solve the problem. Holding the threads seems like the easiest solution to me.