I finally decided to try cutting tissue paper on my Pazzles Vue. I ironed a piece of very thin tissue paper between two pieces of freezer paper, shiny side in, making sure that the tissue paper would not move inside the freezer paper. Then I did a few tiny test cuts to make sure I had the blade depth and pressure settings right for this sandwich. I cut out a 2 inch flower, then carefully peeled the tissue paper away from the freezer paper. It cut very nicely. The tissue paper was attached securely to the freezer paper, but it readily peeled away once I got it started with my poker tool. Give this a try on whatever cutter you have, making sure to do some small test cuts first to get the pressure and blade extension right.
This is the technique that I use to cut fabric that I do not want to stiffen. Since I now know I can cut the thinnest of tissue paper, I am confident that I will be able to cut crepe paper as well. I think I will also try cutting construction paper this way. And while I am at it, I think I will try cutting some felt that is not stiff. Felt makes such a mess of m mats, that I think ironing it in between freezer paper sheets will be not only make the felt cut better, but also keep my mat clear of all of those pesky felt fibers that mess up my mats when I use it.