Emerging three-dimensional printing approaches enable cellular structures with composite walls or struts, creating new opportunities for architected materials with high specific stiffness and damping. This paper describes the damped dynamic response of composite walls comprising elastic and viscoelastic phases, and quantifies the scaling between material and structural loss factors. These descriptions are embedded in a highly efficient finite element framework to predict the steady-state frequency-dependent dynamic response of cellular structures, with an emphasis on structural loss factors arising from viscoelasticity. The framework is then used to compare the response of prismatic structures comprising hexagonal, triangular, rhombic and Kagome cells, and to quantify the scaling relationships between cell properties, structural loss factors and material loss factors. Case studies of structures with heterogeneous cell shapes and composite walls illustrate that significant damping gains are possible through architecture. Increases of a factor of two are possible using simple lattice distortions in a monolithic material, while increases of a factor of ten can be achieved with composite walls when the damping material has a loss factor of  and a modulus of at least 1% of the elastic phase. The results strongly suggest that future topology and materials optimization will enable dramatic gains in damping.