Manganese oxides with the cubic perovskite structure (typified by LaMnO3) have stimulated considerable interest because of their magnetoresistive properties(1-9); they exhibit extremely large changes in electrical resistance in response to applied magnetic fields, a property that is of technological relevance for the development of magnetic memory and switching devices. But for such applications to be viable, great improvements will be needed in both the sensitivity and temperature dependence of the magnetoresistive response. One approach under consideration for optimizing these properties is chemical substitution(10). Here we demonstrate an alternative strategy, in which we synthesize layered variants of the cubic perovskite parent compounds that have a controlled number of MnO2 sheets per unit cell. This strategy is structurally analogous to that employed for the systematic exploration of the high-transition-temperature copper oxide superconductors(11). We find that the magnetoresistive properties of these materials depend sensitively on the dimensionality of the manganese oxide lattice. Although the properties of our materials are still far from optimal, further exploration of this series of layered perovskites may prove fruitful.