We present a new acquisition method that enables high-resolution, fine-detail full reconstruction of the three-dimensional movement and structure of individual human sperm cells swimming freely. We achieve both retrieval of the three-dimensional refractive-index profile of the sperm head, revealing its fine internal organelles and time-varying orientation, and the detailed four-dimensional localization of the thin, highly-dynamic flagellum of the sperm cell. Live human sperm cells were acquired during free swim using a high-speed off-axis holographic system that does not require any moving elements or cell staining. The reconstruction is based solely on the natural movement of the sperm cell and a novel set of algorithms, enabling the detailed four-dimensional recovery. Using this refractive-index imaging approach, we believe that we have detected an area in the cell that is attributed to the centriole. This method has great potential for both biological assays and clinical use of intact sperm cells.