You might try using a pencil erasure to remove those little pieces also. I have used it on threads when I was unstitching (ripping) and it works great.

tissue paper for quilting
 

I have used the blue pen to mark designs and then use a moistened swab to wash off the marks. Also learned from DGD that if you wet paper it tears more easily so would suggest tracing over the stitching with wet swab and it shoud all pull out quickly. Also saw a demo where they stitched without thread over the design on copy paper, (could make several copies at once.) and then dip a foam paint brush into the dust used to pounce on perforated designs. Brush over the punctured paper and you should have your quilting design on your fabric.

What about freezer paper.