Crack propagation in rock-like materials is notoriously difficult to model under the continuum mechanics framework. The phase-field method (PFM) has been emerging as an advantageous tool by treating the crack surface as a continuous function, therefore dealing with a sharp discontinuous displacement field is avoided. In the study, the finite volume method (FVM) has been utilized to discretize the governing equations for the conservation of linear momentum and the evolution of phase field, which are solved by an iterative staggered scheme. Two strain energy decomposition methods, i.e., the spectral decomposition by Miehe et al. [1] and the spherical-deviatoric decomposition by Amor et al. [2], are considered to distinguish the effects of tensile and compressive stresses on crack propagation. The adaptive mesh refinement (AMR) technique is further employed to improve computational efficiency, which is relatively easy to achieve in FVM as the method can naturally handle unstructured meshes with hanging nodes. Several classical crack propagation problems, including the single-edge notched tension and shear tests, the L-shaped panel test, and the test on a notched plate with a hole, are simulated and compared with available experimental and finite element method (FEM) results, which demonstrate that the FVM-based phase-field method can model crack propagation accurately and effectively and could be more efficient than the FEM-based one. Besides, due to the conservative property of FVM, the method could be conveniently coupled with fluid flow to solve multiphysics problems such as hydraulic fracturing.