A methodology for imposing approximate plane strain conditions in magnetic resonance elastography through physical constraint is described. Under plane strain conditions, data acquisition and analysis may be conducted in two dimensions, which reduces imaging and reconstruction time significantly compared with three-dimensional analysis. Simulations and experiments are performed to illustrate the constraint concept. A signal/noise analysis of a two-dimensional linear inversion technique for relative elastic modulus is undertaken, and modifications to the numerical method are described which can reduce the SNR requirements by a factor of two to four. Experimentally measured data are reconstructed to illustrate the performance of the method.