A combination of numerical and modeling approaches is employed to study the effective energy dissipation properties of solid polymer matrixes filled with coated spherical inclusions. For the coating layers, typical viscoelastic properties of a polymer at and well above the glass transition region are assumed. It is shown that by optimizing the thickness of the layers, one can achieve multiphase materials with effective loss characteristics significantly exceeding those of the individual material's constituents. Upon varying the layers' thickness, we detect two distinct peaks of energy dissipation. By analyzing spatial distributions of local energy dissipation rates, we relate the origins of the two peaks to two different states of local strain realizable inside the sections of maximum energy dissipation. We surmise that such lossy composites may prove technologically appealing, as one could use conventional polymer-processing equipment to directly process them into lossy complex-shape structural parts.