Tunneling anisotropic magnetoresistance (TAMR) has been characterized in junctions comprised of face-centered cubic (fcc) Co (111) ferromagnetic electrodes grown epitaxially on sapphire substrates, amorphous AlOx tunnel barriers, and nonmagnetic Al counterelectrodes. Large TAMR ratios have been found, up to ∼7.5% and ∼11% (at 5 K), for the in-plane and out-of-plane magnetization geometry, respectively. Such large TAMR values were not expected a priori, given the weak anisotropy of the (bulk) Co bands due to spin-orbit interaction, and the absence of Co (111) surface states that cross the Fermi energy. Both the in-plane and out-of-plane TAMR effects exhibit a predominantly twofold symmetry, and a strong bias dependence. The in-plane TAMR shows a maximum along the (twofold) magnetic hard axis, suggesting a relation between magnetic anisotropy and TAMR. We propose that uniaxial strain in combination with Bychkov-Rashba spin-orbit interaction, producing an interfacial tunneling DOS that depends on the magnetization direction, is responsible for the TAMR effect. The importance of the interfacial Co/AlOx (electronic) structure for the TAMR effect is underlined by measurements on junctions with overoxidized AlOx barriers, which show markedly different bias and angle dependence.