Thermo-mechanically coupled phenomena are sometimes critical issues in the design of polymeric composite materials. The viscous property in a matrix material results in not only the temperature-dependent mechanical behavior but also the energy dissipation stemming from itself to form the temperature distribution in the microstructure. Since the energy dissipation works as the instantaneous heat source, the unsteadiness emerges in the resulting microscopic heat conduction and, in turn, microscopic mechanical behavior but cannot be embedded into the conventional framework of computational homogenization. We establish a method of two-scale analysis to capture the transient thermo-mechanically coupled behaviors of dissipative composites. Under the assumption of a real-size representative volume element, the approximative two-scale decomposition for a coupled rate potential stored in dissipative composites yields governing equations characterizing the mutual relationship between mechanical equilibrium and unsteady heat conduction in both scales. The key point in the present study is that the periodic boundary condition constraining perturbed temperature is available for dissipative composites, whereas the linear and uniform boundary conditions are only valid in thermo-hyperelastic materials. The results of several numerical examples uncover the capability of the proposed method considering the periodicity of perturbed temperature.