The current article explores 3D bi-material lattice concepts for tailoring compressive stress-strain response. The unit cells of the lattices consist of six stiff inclined struts that together form the edges of two stacked tetrahedra. The central plane (between the two tetrahedra) contains elastomeric elements that stretch when the lattice is compressed. The geometric configurations of the elastomers examined here include: (i) straight struts between each node pair within the central plane, (ii) multiple struts between each node pair, including one straight strut and one or more curved struts, and (iii) flat sheets, either uniform or graded in thickness, connected at the mid-plane nodes. Assessments of the concepts are made using analytical models, finite element simulations, and experiments on lattices fabricated by 3D printing. The experimental results affirm the understanding of mechanical response gleaned from the models and highlight the importance of joint design in attaining large straining capacity and strain reversibility. They also demonstrate a nearly twofold increase in load bearing capacity when straight struts are replaced by graded sheets. The work also raises the prospects for computational design optimization for maximum efficiency in material use.