Markov Random Fields (MRF?s) are an effective way to impose spatial smoothness in computer vision. We describe an application of MRF?s to a non-traditional but important problem in medical imaging: the reconstruction of MR images from raw fourier data. This can be formulated as a linear inverse problem, where the goal is to find a spatially smooth solution while permitting discontinuities. Although it is easy to apply MRF?s to the MR reconstruction problem, the resulting energy minimization problem poses some interesting challenges. It lies outside of the class of energy functions that can be straightforwardlyminimized with graph cuts. We show how graph cuts can nonetheless be adapted to solve this problem, and provide some theoretical analysis of the properties of our algorithm. Experimentally, our method gives very strong performance, with a substantial improvement in SNR when compared with widely-used methods for MR reconstruction.