I've been folding some origami cranes with non-conventional materials. It was challenging, and I definitely need more practice, but it was really fun!
The green and orange cranes are made of polymer clay, the purple crane is made of cold porcelain. I used a small pasta machine to roll thin sheets, that I then cut into squares using a template. I dusted the sheets with cornstarch to prevent them from sticking to the machine, and also to prevent the layers from sticking to each other while folding.
This last crane is made of copper wire mesh. This material was the most difficult to work with. It definitely does not like reverse folds.
Besides paper, what other materials have you used to fold origami models?