Post-It Full Adhesive Notes
I agree with Crafter 4 Life that where you put the tape is critical; keep the adhesive off the "pretty" part of the die-cut.
I was using small pieces of blue painters' tape to hold dies in place, but what I had was stickier than I wanted it to be, and I didn't want to buy a whole roll of the low-tack variety. I tried Post-Its, but it seemed wasteful to use only the sticky area and discard the remaining plain paper.
Then I discovered Post-Its that have adhesive over nearly all of the paper, with only about 3/8" without glue. They're called Full Adhesive Notes, and I found them at Staples. Now I feel better about using small pieces to hold dies in place, and they're working well for me.