We have developed a finite-volume based elasticity recovery method for Magnetic Resonance Elastography. The method accommodates heterogeneity and compressibility, uses only first derivatives, avoids matrix inversion and can be calculated in a highly efficient stencil format. We compared the method to a Helmholtz-type wave inversion. Visually, the method shows sharper rendering of cracks and boundaries. Quantitatively the values of the new method differ from a Helmholtz-type method in proportion to the severity of interfaces, which likely reflects the new method's sharper rendering. The resolution and sensitivity of the method lower the inherent stability, which we address here with a simultaneous multifrequency wave inversion. Future work will introduce sparsity-promoting regularization to deliver both stability and structure.