I've got a huge bag of clippings that I thought I would use to make some doggie beds for the animal shelter down the street. Is there a pattern or certain method that I should check out or is it as simple as sewing 2 squares or rectangles of fabric right sides together and then flipping and stuffing and sewing shut?

Its just that simple. Make them in different sizes. Look on line at the black wire crate sizes to get an idea or ask the shelter what sizes they would like. Some small for the kittys and little dogs and some bigger for the big dogs.

Kimberly what a great idea. I make some for our local shelter. They don't have requirements like hospital quilts or kids charities. They do have some "likes"--they like them to be flat with just an old blanket or towel for batting (no regular quilt batting). They also like them to be a size that's able to fit in their washing machines. Check with the shelter and they'll help you out.
I just sewed together some old samples of men's suiting (OMG some were cashmere). Used an old beach towel for batting, flannel sheet for backing. Sewed it all together and turned it right side out. It was 28 x 28 inches square.
Good luck!

So they prefer flat like a quilt instead of puffy like a doggie bed? I was gonna use my clippings as the stuffing to puff them up a bit. Not a good idea?

I put my clippings in a pillowcase, and then stitch it closed.

Great idea! My dogs love fleece!

This is how I do mine and the shelter was thrilled to get them

I also toss batting, fleece and fabric scraps into an old pillowcase. Once it seems comfy enough, I stitch it closed and send it off the the shelter. They love getting them! :D

I'm thinking all shelters must be different. They love anything you send; I just think if you ask them which they prefer, they'll get more use out the items. I did see a few puffy beds out in the waiting area and in the hallways so they uses both but the flatter ones are easier to wash and dry. Ours can't wash and dry the puffy ones in the machines they have.
Now my dog won't sleep on anything flat--the puffier the better altho he prefers a our bed to anything LOL>

I did that and when it went into the washing machine it almost ruined the machine.