Futon

I skipped the mats as well and built a small platform bed. Be aware the futon mattress are pretty heavy for their size. I still like it lots, although its a queen size and never gets put up, its still beats mattress/spring/ lumps/ frames, etc.

I sleep on a korean yo, which is similar to a shikibuton, but a bit thinner. I live in a small apartment and love having the ability to pack my bed away and have that extra space during the day. It took a little adaptation, but sleeping on a firm surface is now quite comfortable, but not so much so that I'm tempted to sleep in.

I put it directly on a hardwood floor and haven't had any trouble with mold, moisture, or dust, but I do fold it up and put it away most mornings. Leaving it on the floor or another solid surface for a long time would probably lead to the problems you mention.

Since having that floor space is the main attraction for futon-style sleeping, I haven't looked into bedframes. I would think that a slatted bedframe would work as long as the slats were fairly close together. You may also be able to buy tatami bedframes that provide the cushioning and airflow of the traditional tatami mat but with a little elevation.

I did some research into those as well, and to save you some time, J-life's futon and Tuft and Needle's futon are all made by the same factory. I believe it's Gold Bond. I ended up finding the same mattress on Wayfair for much less than the other two sites, and I have been using it for 3 weeks now, no complaints.

http://www.wayfair.com/Gold-Bond-All-Cotton-4-Futon-Mattress-604-GO1093.html?refid=TEM_WF_732&mmid=319017251&csnid=&cltr=