A method for three-dimensional crack propagation by the numerical manifold method (NMM) is analyzed. With the help of mathematical cover and physical cover, no Heaviside function is required to describe a crack in the NMM, and so the three-dimensional crack propagation with NMM can be analyzed being similar to that of two-dimensional cases. Mohr-Coulomb criterion with a tensile cut-off is used to analyze failure state of each crack tip line. A simplified method for determining the final propagating direction is described. According to the final propagating direction, a non-local crack tracking method is used to determine the updated crack tip line. Quadrilateral or triangle tracking method is selected according to the situation during propagation. For boundary faces, in order to make the deformed surface remain plane, new generated faces are needed to be triangulated. Three numerical examples are analyzed to validate the proposed 3D crack propagation method. The numerical results are all in good agreement with those results in the existing studies. Furthermore, a concrete gravity dam under water pressure with an overload factor is simulated and the progressive failure process of the concrete gravity dam is obtained, which demonstrates the applicability of the proposed method in practical engineering.