The problem with your requirements (no box spring, no metal, no wood) is that it will be nearly impossible to support your mattress without a frame and supports (metal or wood).
You see, anything you stack, like patio paving stones, for example, will only support the edges of the mattress. So, when you get into bed, the mattress will concave in the middle, like a curled potato chip. And, because you don't want to use any wood, you are suggesting to eliminate supporting the mattress.
I suppose you could create a rope hanger, much like the rope hanger designs used to hang potted plants, and support it from the ceiling with some heavy duty hardware...but hardware is metal.
Without the use of a boxspring, wood or metal, you are eliminating 99.9% of your options.