Thin liquid films flowing down vertical or inclined plates are widely used in many industrial apparatuses. Using structured plate surfaces often leads to heat transfer enhancement. In the present work, a numerical model for heat transfer in a thin liquid film flowing down a heated, inclined, and grooved plate is developed. To this end, the Graetz-Nusselt problem for falling films on structured plates is solved. The computed velocity field and the developed temperature field, as well as the temperature distribution in the thermal entrance region of a falling film, are presented. The dependence of the temperature distribution on the Reynolds number, Biot number, applied heat flux, plate inclination angle, and plate topography is investigated. It is shown that the film rupture on the groove crests, which is observed in experiments at relatively high heat fluxes, can be attributed to the strong interface temperature gradients developing in the thermal entrance region. To qualitatively validate the numerical model, the hydrodynamics and heat transfer in falling films on structured plates are studied experimentally and simulated using the CFD tool FLUENT. The numerical results of the Graetz-Nusselt problem are discussed and compared with the experimental values and results of the FLUENT simulations.