I am currently working on a quilt for a friend ... one part of the quilt is silhouette of dog breeds hand appliqued onto blocks .....not sure if this is something you might consider?

BQ or BQ2 patterns would really highlight the fabric you have chosen. You can google for images.

How about snowball block with a nine patch made of plain and fabric with pets on.

I recently purchased a Log Cabin book that had a wall hanging project in it which used a center square of fabric with several narrow borders Log Cabin style (1" finished) in several different colors. They used a toile for the center, but I think this fabric would work great for the center.

You could do large blocks, 9 or 12 inches in the fabric you have in the photo and alternate it with a blocks of a silhouette of a cat and dog (get the outline from a colouring book), do the silhouette in black and appliqué it. Another block could be an appliquéd heart. The red and white appliqués would look nice on top of your featured fabric

the first thing that popped into my head was to cut this out as a puppy and kitty and applique it on black or red fabric. Then use a variety of small animal prints to frame them. But, any block that highlights this as the center would also work great.

I like Warm Wishes for fabric like this. http://www.ashevillequiltguild.org/A...or%20Linus.pdf
I have been cutting my squares 7 1/2"and the strips 3 1/2" for the center one and 2 1/2" each for the side ones. Makes a 7" block so you read the cute fabric.

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.