A finite volume method (FVM) is developed for elastic analysis of three-dimensional (3D) anisotropic linear elastic solids with continuously variable material properties. The method employs tetrahedral elements so that it owns adaptability to problems with irregular domain. A staggered unstructured FVM is used for spatial terms. The displacement, velocity and acceleration are defined on the cell vertex while the stress and material properties which are uniform in the cell are defined on the cell center. An Euler scheme is employed for time dependent terms. The numerical results of the FVM agree well with the results of other numerical methods, and the FVM consumes much less computational time and memory, which is able to predict the 3D anisotropic linear elastic problems accurately and efficiently.