We report giant magnetoresistance in [Ni81Fe19/Cu] N multilayers. Saturation magnetoresistance values exceeding 16% for saturation fields of only 600 Oe are found at 300 K. In addition, we show evidence for well-defined oscillations in saturation magnetoresistance as a function of Cu spacer layer thickness at 4.2 K, with an oscillation period and phase similar to that in Co/Cu multilayers. However, the temperature dependence of the magnetoresistance, while weak for thin Cu layers, is much stronger for thicker Cu layers. Consequently at 300 K only a single oscillation in magnetoresistance for thin Cu layers is found. We show that the properties of the Ni81Fe19/Cu multilayers are very sensitive to annealing at moderate temperatures, which may limit the possible technological applications of such structures.