This study aims to develop a framework called MP-Dixon-GRASP (Magnetization-Prepared Dixon-based Golden-angle RAdial Sparse Parallel MRI) for rapid free-breathing fat/water-separated 3D T1 mapping of the liver. The technique combines inversion recovery-prepared multiecho stack-of-stars acquisition with subspace-based sparse image reconstruction. The performance of MP-Dixon-GRASP was evaluated in fat/water phantoms and in subjects with normal and elevated liver fat content. The results suggested that fat/water-separated T1 mapping is able to remove the influence of fat, which enables more accurate estimation of true T1 values in the liver. With fat/water-separated T1 estimation, MP-Dixon-GRASP could be potentially useful for imaging patients with fatty-liver diseases.