We present an acquisition method for optical coherence elastography (OCE) that enables acquisition of three-dimensional elastograms in 5 s, an order of magnitude faster than previously reported. In this method, based on compression elastography, the mechanical load applied to the sample is altered between acquisitions of consecutive optical coherence tomography volume scans (C-scans). The voxel-by-voxel phase difference between the volumes is used to determine the axial displacement and determining the gradient of the axial displacement versus depth gives the local axial strain. We demonstrate sub-100-microstrain sensitivity and high contrast in elastograms, acquired in 5 s, of structured phantoms and freshly excised rat muscle tissue that are comparable in strain sensitivity and dynamic range to our previously reported B-scan-based method. The much higher acquisition speed may expedite the translation of OCE to clinical and in vivo applications.