Thank you for all the advice you left in the comments on my applique post of last week. Clear thread - that's the answer! I even had some on hand, bought several years ago at a quilt show. So now I'm using Superior Thread Mono Poly and am very happy with the results. It is a very fine, clear polyester monofilament thread.
I put the thread on my machine then played around with some different stitches, stitch lengths and widths, and thread tension, until I found the combination I like best. I'm using the hem stitch (#7 on Bernina 440Q) with stitch width of 2 and length of 1 and I loosened the thread tension so the bobbin thread doesn't get pulled up to the front. I have ecru Aurifil thread in the bobbin.
So far, so good, and I'm well on my way with the blocks. I'm on the lookout for some dark brown batik fabric for the body section and may use matching thread for that work.