The ballistic magnetoresistance of tunnel junctions that comprise Mn films is found to exhibit oscillations with increasing Mn-film thickness, as is investigated by means of first-principles electronic-structure and transport calculations. The period of two monolayers is directly related to the layer-wise antiferromagnetic structure of the Mn films, in particular to the alternating magnetization at the interfaces. These findings substantiate unequivocally the effect of the electronic and magnetic structure of interfaces on the conductance of tunnel junctions.