This is a really easy, cost effective way to get all sorts of great paper. I just take regular computer printer paper, a sponge, and artist's ink, and you can get all sorts of fantastic papers. I would recommend using ink over paint, because heavier bodied paints will make the paper stiffer (and harder to fold), while watercolor will have a tendency to make computer paper wrinkled. I had ink already, but you can get bottles that will last you a long, long time from any art store..
I also have an art sponge, with fine pores, but any tool you have for moving the color onto the paper will work. I simply drip a little ink directly onto the sponge, and then dab, push, or smear it into the paper. Don't worry too much if you have a few spots that pick up too much ink, or don't have enough, or otherwise look a bit odd; once you've folded it into your origami model, those flaws probably won't be very noticable. They might even be attractive.
I suggest using two colors at a time, on different sides of the brush/sponge. Since orange is one of my primary colors, I've been using yellow and red, and letting them blend in the middle of the sponge. This gives me a fast range of colors in a single swipe. It can tend to be a little bit less perfect looking, but for this look, you're not really after perfection. The messiness looks good.
I will suggest, though, being careful about trying to blend black in with colors, or two colors that are complementary (like blue and orange, or red and green). While not entirely unattractive, you'll probably wind up with something more brown/gray in those mixes. But if you wind up with something you don't like, don't feel bad about throwing out a piece of paper. When you've finished painting your sheets, just cut them into squares and fold into your origami!
(I plan to revamp this post later when I can get some good photos taken. It'll probably more graphic heavy and less chatty then.)