Rugs that don`t damage laminate floors

I think I have been lucky. Just discovered that the backing on my turtle mat rugs is degrading, carried one rolled up rug and my hands got covered in black dots from the backing. It has been stored for a few months, never damp. Now I read that rubber and latex backing on rugs can damage and/or discolour even expensive laminate boards. All 4 rugs have de-graded backing. I need rugs, will have to get some new ones for downstairs. Will be living on a building site for many more months. No idea what type. Does anyone know?