We use extensive molecular dynamics simulations to calculate the thermal conductivity and thermal diffusivity in three common molten salts, LiF, LiCl, and KCl. Our analysis includes the total thermal conductivity and intrinsic conductivity, excluding mass currents, measured experimentally. The latter shows good qualitative agreement with the experimental data. We also calculate their key thermodynamic properties, such as constant-pressure and constant-volume specific heats. We subsequently compare the results to the lower bound for thermal diffusivity expressed in terms of fundamental physical constants. Using this comparison and recent theoretical insights into thermodynamic and transport properties in liquids, we interpret thermal properties on the basis of atomistic dynamics and phonon excitations. We finally find that the thermal diffusivity of molten salts is close to their kinematic viscosity.