The paper is concerned with a low-order finite element method, namely the staggered cell-centered finite element method (SC-FEM), which has been proposed and analyzed in [1] for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes, including a dual mesh and its tetrahedral submesh from a general primal mesh. The displacement is approximated by piecewise trilinear functions on the subdual mesh and the pressure by piecewise constant functions on the dual mesh. As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types.