We study wave propagation in periodic materials with dissipation using two different formulations. An -formulation yields complex frequency solutions for nonvanishing dissipation whereas a -formulation leads to complex wave numbers. For small (realistic) levels of material dissipation and longer wavelengths, we show that the two formulations produce nearly identical results in terms of propagation constant and wave decay. We use the -formulation to compute loss factors with dissipative bandgap materials for steady-state wave propagation and create simplified diagrams that unify the spatial loss factor from dissipative and bandgap effects. Additionally, we demonstrate the applicability of the -formulation for the computation of the band diagram for viscoelastic composites and compare the computed loss factors for low frequency wave propagation to existing results based on quasi-static homogenization theory.