We propose a novel sparse dictionary learning frame work to recover dynamic images from under-sampled measurements. Unlike the recent low rank schemes, the proposed scheme models the dynamic signal as a sparse linear combination of temporal basis functions chosen from a large dictionary. Both the basis functions and the sparse coefficients are estimated from the undersampled data. We show that this representation is much more compact compared to the low rank models. We also develop an efficient majorize-minimize algorithm to estimate the sparse model coefficients and the dictionary directly from the measured data. We compare the proposed scheme against low rank models and compressed sensing, and demonstrate improved reconstructions in the context of myocardial perfusion imaging in the presence of motion.