Thermal transport through model copper-polyethylene interfaces is studied using two-temperature nonequilibrium molecular dynamics. This approach treats electronic and phonon contributions to the thermal transport in the metallic region, but only phonon mediated transport is assumed in the polymer. Results are compared with nonequilibrium molecular dynamics simulations of heat transport in which only phonon contributions are incorporated. The influence of the phase of the polymer component (crystalline, amorphous, and lamella) and, where relevant, its orientation relative to the metallic interface structure is explored. These computational studies suggest that the thermal conductivity of the metal-polymer interface can be more than 40 times greater when the polymer chains of the lamella are oriented perpendicular to the interface than the situation when the interface is formed by an amorphous polymer or a crystalline polymer phase in which the chains orient parallel to the interface. The simulations suggest that the phonon contribution to the thermal conductivity of the copper region can be increased by as much as a factor of three when coupling between the electrons and phonons in the metal region is incorporated. This, combined with the explicit inclusion of the purely electronic component of the thermal transport in the metal region, can lead to a substantial increase in the heat flux promoted by the interface while maintaining a constant temperature drop. These simulation results have important implications for designing materials that have excellent electrical insulation properties but can also be highly effective in heat conduction.