Cutting off the corners never worked for me either.
For pieces 1 yard or more in length, I would accordion fold (about 12" per fold) and then safety pin the edges together.
For smaller pieces (1/2 yard or so) I would invest in some mesh bags and put one in each bag. (Check the Dollar Store first for these.)
For fat quarter sized pieces I would be tempted to use mesh bra containers like this one:
http://www.amazon.com/Woolite-W-8243...dp/B004K6L83I/
although I bought mine at Walmart. They are great for machine washing bras and panties, and I think would work well for a fat quarter in each side.
You could probably save money by making your own mesh bags -- purchase a couple of yards of mesh, wash it (should not ravel the way cotton fabrics do), then sew bags. Instead of zippers I would just make a 3-layer bag (similar to how one would make a pillow cover without a zipper, overlapping the flaps on the back) and safety pin the mesh bag closed before tossing in the wash.
The mesh bags take up very little storage room and can be used for future fabric purchases too!