Compressed-sensing reconstruction using motion estimation and compensation for dynamic MRI data is proposed. Reconstruction is driven from a residual in the k-space domain between the current-frame measurements and a corresponding motion-compensated prediction. Due to the periodicity commonly exhibited in dynamic MRI, a telescopic motion search through the entire group of pictures is used to determine the best match for the block-based motion estimation. Experimental comparisons demonstrate improved performance as compared to existing dynamic-MRI reconstructions, both those with and without motion compensation.