Apparatus and methods for cutting compressible cellular polymers such as polyurethane foam, are characterized by a first die roller and a pressure roller, wherein both rollers define raised portions separated by recesses or depressions, and the raised portions on the first die roller register with the recesses of the pressure roller at a nip formed between the rollers. Portions of the polymer material extruded into the spaces between the raised portions of the first die roller are cut away, such that the cut product conforms substantially to the surface geometry of the first die roller, and the depth of cut is greater than with prior rotary cutting methods.