The role of localized phonon modes in phonon-assisted tunneling in GaAs/AlAs double-barrier resonant tunneling structures is considered for a range of temperatures and magnetic fields. Phonon modes are calculated using a dielectric continuum model and electron-phonon Hamiltonians are presented for the most important modes. Formulas for phonon-assisted tunneling currents are derived that express the inherently three-dimensional process in a simple one-dimensional form. It is found that the excess current due to phonon-assisted tunneling in typical structures is caused primarily by two types of localized modes: confined modes in the well and symmetric interface modes, with interface modes dominating in structures with narrow wells. Current peaks broaden with increasing temperature, and for temperatures ≳20 K the resolution of features due to distinct phonon types is very difficult. The application of a magnetic field parallel to the current flow leads to a complex spectrum of sharp current peaks corresponding to various inter-Landau-level transitions which occur during phonon-assisted tunneling.