This paper compares the accuracy of elastic and elastic-plastic solid continuum finite element analyses modeled with either all hexagonal or all tetrahedral meshes. Eigenvalues of element stiffness matrices, linear static displacements and stresses, dynamic modal frequencies, and plastic flow values are computed and compared. Elements with both linear and quadratic displacement functions are evaluated. Linear incompressibility conditions are also investigated. A simple bar with a rectangular cross-section, fixed at one end, is modeled and results are compared to known analytical solutions wherever possible. The evaluation substantiates a strong preference for linear displacement hexagonal finite elements when compared solely to linear tetrahedral finite elements. The use of quadratic displacement formulated finite elements significantly improve the performance of the tetrahedral as well as the hexahedral elements. The nonlinear elasticplastic comparison indicates that linear hexagonal elements may be superior to even quadratic tetrahedrons when shear stress in dominant. Results of this work may serve as a guide in selecting appropriate finite element types to be used in three dimensional elastic and elastic-plastic analysis.