The homogenization of viscoelastic composite materials gives rise to long-memory effects, reflected by the presence in the homogenized viscoelastic constitutive law of a delayed stress response expressed as a convolution integral involving the past material strain history. Computations reveal that the long-memory has the ability to delay or even absorb waves depending on the fraction of reinforcement and relative properties of the building phases with the composite. An approximation by a one-term Prony series of the computed spectrum of retardation times is shown to provide an accurate representation of the continuous spectrum of retardation times when simulating creep experiments. The long-memory effects entail shifts of the frequency band diagrams for both the natural and damped frequencies due to the presence of damping. Decreasing the effective retardation time of the composite leads to a faster attenuation of propagating longitudinal and shear waves.