The paper investigates free and forced vibrations of structurally inhomogeneous viscoelastic rod mechanical systems. A mechanical system is considered in which the rheological properties of the deformable elements differ significantly: some elements are elastic, while others are viscoelastic with different hereditary functions. Massive deformable elements have finite volumes, whereas massless elements have finite or negligibly small volumes. The deformable elements of the system are made of viscoelastic materials, such as polymers and polymer-based composites, whose physical properties are described by linear Boltzmann–Volterra hereditary constitutive relations with integral difference kernels. The main objective of the work is to study the dissipative properties of such structurally inhomogeneous rod mechanical systems. Moreover, in free oscillations of the system, the manifestation of dissipation reduces to the attenuation of oscillations, the attenuation rate quantitatively assesses the dissipative properties of the system; in steady-state forced oscillations, the dissipative properties are most pronounced in resonant modes and lead to finite values of resonant amplitudes. The natural frequencies and forms of oscillations are determined from the condition that the determinant of the system, calculated by the Müller-Gauss method, is equal to zero. For the considered mechanical system, the fundamental possibility of significantly intensifying dissipative processes in dynamical systems and reducing the resonant amplitudes of principal oscillations due to the convergence of corresponding natural frequencies is shown.