Via Lifehacker, I learned about Furoshiki, a traditional Japanese cloth wrapping. You’re less likely to get paper cuts than with origami, but you still get the fun schematic diagrams.
Apparently the Japanese Environment minister (they have an environment minister? Is that essentially the Dept. of the Interior?) wants to promote the use of them. Although you can buy them from vendors, it’s the sort of thing that screams “DIY.”