This work examines the generation of heat in silicon MOSFETs using self-consistent Monte Carlo device simulation with full electron bandstructure and a full phonon dispersion computed from the Adiabatic Bond Charge model. We devise an efficient algorithm for the inclusion of full phonon dispersion in order to account for anisotropy and details of heat transport with great accuracy. We compute the density-of-states (DOS) and the lattice thermal energy numerically and use them to generate maps of local temperatures in a representative small-channel MOSFET device. Our results show that most heat is dissipated in the form of optical g-type phonons in a small region in the drain, and that the heat flows in a preferred direction aligned with the flow of the electron current. We also show that the distribution of generated phonons in energy closely follows the phonon DOS.