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A new strain energy-based method for homogenization of 2-D and 3-D cellular materials is developed using an extended version of Hill's Lemma for the non-Cauchy continuum satisfying the micropolar elasticity theory. Both kinematic and mixed boundary conditions (BCs) that obey the Hill-Mandel condition are identified. The newly proposed method requires that the nodal equilibrium equations be explicitly satisfied at all boundary and interior nodes, unlike the existing approach which does not enforce the equilibrium conditions at interior nodes. For 2-D cellular materials, it is shown that purely kinematic BCs (with prescribed displacement and micro-rotation vectors) and mixed BCs (with prescribed couple stress traction and displacement vectors) can both be used for homogenization, and periodic constraints can be readily accommodated in the former. For 3-D cellular materials, it is demonstrated that mixed BCs (with prescribed couple stress traction and displacement vectors) enable the homogenization satisfying the Hill-Mandel condition. Two sample cases of homogenization of cellular materials are studied by applying the new method – one for a 2-D lattice structure and the other for a 3-D pentamode material. For the 2-D lattice material, the effective classical and micropolar stiffness tensors are obtained using purely kinematic BCs (with and without periodic constraints) and mixed BCs. The closed-form expressions of the effective stiffness tensor components derived here are compared to those given by the existing strain energy-based homogenization method and are seen to be more accurate. For the 3-D pentamode material, mixed BCs are employed in the homogenization, and the effective stiffness tensor components are obtained in closed-form expressions for the first time. The numerical results for the pentamode material given by these analytical formulas are found to agree well with those provided by a finite element model constructed using COMSOL. [ABSTRACT FROM AUTHOR]