Mechanical metamaterials are those of mechanical properties unachievable or beyond those in conventional properties. Typical examples among them are those of negative Poisson's ratio (auxeticity) and/or negative thermal expansion coefficient. In the present contribution, a class of 2D/3D lattice structures with tunable auxeticity, negative thermal expansion and degree of elastic anisotropy are designed. Their effective properties are evaluated by analytical and computational homogenization methods. The tunable range of their effective Young's modulus, Poisson's ratio, and thermal expansion coefficient are explored. Due to their novel properties, they can be widely applied in engineering systems for structural or functional purposes at the macro and micro scales. For instances, the lattice structures may sever as the electrode in lithium-ion battery to enhance the electrochemical performance.