Here's the scoop, if anyone's interested:
My machine, the Sizzix BigKick, works with lots of different dies (including other companies').
Sizzix Sizzlets do NOT cut through fabric.
Sizzix Originals DO! (Jumping Jacks, Hoots, Hollers!)
I've heard that other dies work also, including Sizzix Bigz. But I'm just telling you what I know first hand. I've had good luck finding Sizzix Originals dies at good prices through ebay.
Just thought I'd pass all this on, since it took me forever and a day to figure it all out myself! Happy (and fast!) cutting!