We study the dynamics of polymers in a random disordered medium of fixed obstacles using kinetic Monte Carlo methods. The polymers can have monomers which have attractive (A-type), repulsive (R-type) or neutral (H-type) interactions with the fixed obstacles that comprise the disordered medium. Several classes of homopolymers and heteropolymers with diverse sequences have been studied. Our most noteworthy result is that, above a threshold temperature, polymer bearing monomers that are attracted to the disordered medium translocate faster through the medium than those bearing neutral or repulsive monomers. We discuss how a delicate balance between energetic and entropic factors leads to this counterintuitive outcome. By examining heteropolymers with different sequences, we also find that the dependence of mobility on average composition is stronger than that on higher order correlations characterizing the sequence distribution. Connections between our results and experiments with synthetic and biological systems are noted.