The ballistic conductance of Fe/MgO/Fe magnetic tunnel junctions depends significantly on the direction of the magnetization in the leads, as is investigated by relativistic first-principles electronic structure and transport calculations. The rotation of the parallel aligned lead magnetizations from in-plane to perpendicular orientation with respect to the interfaces, that is tunnel anisotropic magnetoresistance (TAMR), increases the zero-bias conductance by about 30%. The effect originates from both the Rashba spin-orbit interaction at the interfaces and from resonant tunneling. Spin-orbit induced band gaps in the leads show no considerable effect of the anisotropic magnetoresistance. The tunnel magnetoresistance (TMR), i. e. the dependence of the conductance on the mutual angle between the lead magnetizations, is also addressed.