I cut my black background squares last night, and started sewing today, following the tutorial that I linked to earlier. What I discovered was that, sewing by hand (I don't presently have a functioning machine) resulted in some really ugly squares.
So I decided to take a different tack - literally. I had already marked 1/4" from the edge on my squares, for a sewing line. I decided to use that as a line on which to fold and press the edges. Then I folded the square in half, first one way and then the other, finger-pressing to mark the center of the square. Next, I folded each corner toward the center, hiding the folded raw edges. I tacked all the corners down in the center, and pressed all the folds. Then I folded the corners to the center again, again pressing and tacking them down.
Now I have some much-better-looking squares. I've started whip-stitching the squares together, and I've cut the batik squares for the "windows". I'll scan some to show you when I get some more put together.