A new kinetic theory approach for calculating the thermal conductivity of a dilute polyatomic gas from the intermolecular pair potential is presented. The contributions due to internal degrees of freedom have been separated into a classical rotational and a quantum-mechanical vibrational part. Assuming that the vibrational states of the molecules do not significantly influence the collision trajectories, and that vibrationally inelastic and vibrationally resonant collisions are rare, we have obtained a simple self-diffusion mechanism for the vibrational contribution to the thermal conductivity. For non-polar gases like methane or nitrogen, the new approach yields thermal conductivity values that are very close to those obtained with the previously used kinetic theory approach. However, for polar gases like hydrogen sulphide and water vapour, the values obtained with the new scheme are much closer to the most accurate experimental data.