We have investigated the electronic transport and the anisotropic magnetoresistance in systems consisting of pairs of antiferromagnetically aligned layers separated by a non-magnetic layer, across which an antiferromagnetic coupling between the double layers is established. Calculations have been performed within the framework of the tight-binding model, taking into account the exchange coupling within the ferromagnetic layers and the Rashba spin-orbit interaction. Conductivities have been evaluated in the ballistic regime, based on Kubo formula. We have systematically studied the dependence of the conductivity and of the anisotropic magnetoresistance on several material and structural parameters, such as the orientation of the magnetic moments relative to the crystalline axis, band filling, out-of-plane hopping and spin-orbit parameter.