Free oscillations of a one-dimensional Cosserat continuum model are considered. The model is built on the base of Ilyuishin’s mechanical modeling approach. It consists of a beam supplied by rigid massive inclusions periodically placed along the longitudinal line of the beam. Those inclusions are connected with their nearest neighbors by belt drives. We consider bending-tension motion of this construction in one plane. Model behavior in bending and tension motions of the supporting beam and relative rotation motions of the inclusions is elastic. Model behavior in the moment action of inclusions on supporting beam elements is viscoelastic (Kelvin-Voigt model is used). The linearization of the such model motion equations is made for the case of small departures from undeformed configuration. The problem of free oscillations is considered for the linearized model with boundary conditions as follows: pinning of the beam’s edges and absence of moment actions on the end inclusions. The fundamental difference between the system of equations for this model and for the one with fully elastic behavior is mentioned. The general solution of the problem of free oscillations is examined on the assumption with the special form of solution. The “antenna type” construction with known stress-strain properties is taken as an example. The computational solution of the problem of free oscillation is obtained for such construction. It is found that for each oscillation mode there exist exactly two forms of motion. The rate of decay became apparent to depend on viscosity value and oscillation mode. Some graphs are given to demonstrate the dependence.