In this paper, we investigate the use of a consecutive-interpolation for polyhedral finite element method (CIPFEM) in the analysis of three-dimensional solid mechanics problems. A displacement-based Galerkin weak form is used, in which the nodal degrees of freedom (DOF) and their derivatives are both considered for the approximation scheme. Based on arbitrary star-convex polyhedral elements using piecewise linear shape function, the present method can have the advantage of being applicable to complicated structures. Nevertheless, the proposed interpolation technique gives higher-order continuity, greater accuracy with the same number of DOFs. The reliability and efficiency of the CIPFEM are proved by comparing the present results with those obtained by the consecutive-interpolation for tetrahedral element (CT4), conventional linear FEM using polyhedral elements (PFEM), and tetrahedral elements (T4) through numerical examples. Cantilever beam, concrete corbel, and complex hollow concrete revetment block are considered to show the excellent performance of the present approach.