I am making my first set of placemats and would like to know what I can use for batting that is relatively inexpensive but is durable and washable??

Warm and Natural which I buy with my coupon from JoannFabrics You can get a lot of placemats out of one yd (metre) of batt.

If money is really tight you can even use the best parts of a worn beach towel, bath towel or blanket. My grandma and mom used to use anything they had. Also used baby receiving blankets are good too. As long as they've been washed there won't be any shrinkage and they're very washable.
If you use a towel cut away the part about 3 inches in from the end where that "ribbing" is. You don't want to use that.

I'd just use interfacing for a placemat if you want them to really lay flat. That's what I've used and I haven't had any problems.

I just use old sheets or towels it works great.

i agree with mic-pa warm and natural is the way to go

Ditto

Me too!

Or pieces from a worn out mattress cover. The centers wear out, but the edges are still good. They work great for placemat batting. I also use it when quilting a pillow top. You don't need backing for the top when you use a piece of mattress cover, since the backing is inside the pillow anyway.

Thanks for your advice everyone. I have most of the suggestions and will choose one. Thanks