The effects of the spin-orbit coupling (SOC) on the tunneling magnetoresistance of ferromagnet/semiconductor/normal-metal tunnel junctions are investigated. Analytical expressions for the tunneling anisotropic magnetoresistance (TAMR) are derived within an approximation in which the dependence of the magnetoresistance on the magnetization orientation in the ferromagnet originates from the interference between Bychkov-Rashba and Dresselhaus SOCs that appear at junction interfaces and in the tunneling region. We also investigate the TAMR effect in ferromagnet/semiconductor/ferromagnet tunnel junctions. The conventional tunneling magnetoresistance (TMR) measures the difference between the magnetoresistance in parallel and antiparallel configurations. We show that in ferromagnet/semiconductor/ferromagnet heterostructures, because of the SOC effects, the conventional TMR becomes anisotropic-we refer to it as the anisotropic tunneling magnetoresistance (ATMR). The ATMR describes the changes in the TMR when the axis along which the parallel and antiparallel configurations are defined is rotated with respect to a crystallographic reference axis. Within the proposed model, depending on the magnetization directions in the ferromagnets, the interplay of Bychkov-Rashba and Dresselhaus SOCs produces differences between the rates of transmitted and reflected spins at the ferromagnet/semiconductor interfaces, which results in an anisotropic local density of states at the Fermi surface and in the TAMR and ATMR effects. Model calculations for Fe/GaAs/Fe tunnel junctions are presented. Finally, based on rather general symmetry considerations, we deduce the form of the magnetoresistance dependence on the absolute orientations of the magnetizations in the ferromagnets.