Mapping the longitudinal relaxation time constant (T1) of the myocardium using Magnetic Resonance Imaging (MRI) is an emerging technique for quantitative assessment of the morphology and viability of the myocardium. However, three-dimensional (3D) T1 mapping of the heart is challenging due to the high dimensionality of the signal and the presence of cardiac and respiratory motions. We propose a subspace-based method for free-breathing 3D T1 mapping of the heart without respiratory gating. The image function is represented as a high-order partially separable (PS) function to explore the inherent spatiotemporal correlations of the underlying signal. A special data acquisition scheme enabled by the high-order PS model is used for sparse sampling of the (k,t)-space, where complementary sparse datasets are acquired, each covering only a small portion of the (k,t)-space to characterize a single subspace (spatial or temporal). High-resolution dynamic MR images are reconstructed from the highly undersampled (k,t)-space using low-rank tensor and sparsity constraints. We demonstrate the feasibility of our proposed method using in vivo data obtained from healthy subjects on a 3T MR scanner. The proposed method can enable new clinical applications of T1 mapping in cardiac MR.