Another idea is to apply FABRIC Mod Podge to the back of the fabric- just a light layer. It stops it from fraying and still leaves a soft hand. It's usually done for fabric applique, but don't see why it wouldn't work on that fabric. You let it air dry for 45 minutes, then use. Only the FABRIC one will work- any other kind will turn your fabric into vinyl!

I have had regular cotton quilting fabric fray very badly. Later, I realized that the weave was way too loose. I tamed the problem by sewing 2 seams each time. One was the regular 1/4" seam and the next was a 1/8" from the outside. I, also, shortened up the stitch length and I always use a 50 wt thread. I made sure the block was going to be the right size before I sewed the second 1/8" stitch. I did not make sure I stayed a true 1/8" because I wanted it to vary so I didn't loose any more threads from the fabric. My machine has an over edge stitch similar to a serger, but the fraying stopped with the second seam.

I agree with the lightweight stabilizers.

Why on earth would you agree to work with polyester? You won't be able to press those non-cotton fabrics the same as the cottons. Good luck.

Let me be the voice of experience for you here! I used one fabric in a quilt for my daughter that seemed inclined to fray when I was working with it. Later on, every seam with that fabric in it frayed past the seam line in the quilt and was a big mess! I had to hand sew repairs all over the quilt where that fabric was.
Luckily it was a quilt for family and not one that was gifted out!!! I would definitely either use fray stop or some other type of treatment on the edges, or I would iron it to a backing of some sort. In some places that fabric frayed out over 3/8".
Now if I have fabric that seems to fray, either I don't use it, or I seal the edges. By the way, my fabric was also cotton, but not tightly woven enough.

All you have to do is use a short stitch, and stitch around anything you fear might fray. I do this when I get a quilt top dirty and must wash it before I iron it...I use glue sticks a lot and any excess glue just draws dirt..I',m afraid to iron the top for fear of setting in the stains.....but if you stitch around it, it won't ever fray.

In addition to using a light weight stabilizer to prevent raveling, you might try sewing a straight line around the pieces. I know that is a lot of extra work, however, when I first started sewing my clothing and the fabrics back then raveled a lot, so I sewed around each piece, which kept the fabric from fraying, as the stitching stopped the fraying when it met the stitching. I'm looking forward to seeing the finished quilt.

Sounds like this will be a learning experience.

If you have any scraps, use them to try out these treatments.

Thank you all so much for the wonderful suggestions. I have pinking shears and iron-on stabilizer and can get some Fray Block at my local Joann's. I think it will be worthwhile watching the Fons and Porter video first, then doing some experimenting to see which will work best for me. It occurred to me in the middle of the night last night (as most of the most interesting thoughts do) that even if I got these silky pieces sewn into the blocks, and I was confident they wouldn't fray on the inside over time, I could get to the quilting part of the process only to find that quilting on this fabrics pulls threads or whatever, so I'll make a few practice blocks and quilt them before I get serious putting this thing together. Thank you again for all the great suggestions. If I remember, I'll come back and let you know how this experimenting worked out.

I agreed because my daughter loves texture and wanted some in her quilt. When she asked I told her "you pick it out and I'll use it." We did throw one choice out because it wouldn't stop bleeding in the prewash, even after about 10 soakings in a bucket of water. I made sure everything she chose holds up under a cotton setting on my iron as well as washing in hot water and going through a hot dryer. I think some of the non-cotton fabrics are pretty sturdy these days. A couple of her choices looked like some kind of faux suede, and I must admit these odd fabrics do look beautiful in the layout stage so far.