In this study, the effect of energy dissipation on harmonic waves propagating in one-dimensional monoatomic linear viscoelastic mass-spring chains is investigated. In particular, first dispersion laws in terms of wavenumber, attenuation, and wave propagation velocities (phase, group, and energy) for a generic viscoelastic mass-spring chain are derived from the homologous linear elastic (LE) expressions in force of the correspondence principle. A new formula for the energy velocity is introduced to account for energy dissipation. Next, such relations are specified for the Kelvin Voigt (KV) and the standard linear solid (SLS) rheological models. The analysis of the KV mass-spring chain in the high-frequency regime proves that the so-called wavenumber-gap is not related to energy dissipation, as assumed in previous studies, but is due to the nonphysical rigid behavior of the model at high frequencies. The SLS mass-spring chain, in fact, does not show any wavenumber-gap and at high frequencies recovers the wavenumber dispersion curve of the LE system. The behavior of the energy velocity for the different mass-spring chains confirms this conclusion.