There are many easy and fun quilt blocks that are pieced and look like a cat or a dog, dog bone, paw print.
Google dog quilt blocks or cat quilt blocks.
If you think you might make more than one quilt with this theme I highly recommend Janet Kime's books. Either The Cats Meow or Christmas Cats and Dogs. Both have simple blocks that look like cats and dogs but the Christmas one has more dogs the Cats Meow only has one doggy pattern and also a fire hydrant. You can pick either of them up used for $3 plus shipping on Amazon.
You can alternate cat blocks, dog blocks and plain blocks of your rescue fabric for a very cute wallhanging.