A mixed variational principle and derivation of two simple and efficient tetrahedral finite elements with rotational degrees of freedom (DOF) are presented. Each element has four nodes. Every node has six DOF, which include three translational and three rotational DOF. Each element is capable of providing six rigid-body modes. The rotational DOF are based on the displacement formulation, while the translational DOF are hinged on the hybrid strain Hellinger–Reissner functional. Explicit expressions for stiffness matrices are obtained. Element performance has been evaluated with benchmark problems, indicating that they have superior accuracy compared with other lower-order tetrahedral elements.