I've used the wet Q-tip method successfully. I also find that tugging on opposite sides of the block and twisting it a bit makes the paper tear along the seams, which makes removal much easier. I would start here I'd you're really worried about the fabric running, as it avoids water.
The method that looks easiest, but terrifies me the most, is what MarionsQuilts describes here:
Best way to remove paper piecing paper!. She put the whole top in the washing machine, and it worked like magic! I'm not brave enough to try this yet, but I have that same JN snowflake kit sitting on my sewing desk waiting for me, and I would really like a quick way of removing the papers. Whatever method you go with, please let us know how it works!