Recent studies show that the anisotropic behavior of frictional forces is an important factor in contact problems. This research provides the detailed explanation of dynamic behavior of a mass point and thin plates with circular and elliptical contact areas on the rough surface. Symmetric and asymmetric orthotropic frictional forces were mainly studied. The presented model describes terminal movement of bodies in the horizontal plane. Some situations are analyzed numerically for regular pressure distribution. Results for the asymmetric orthotropic friction for elliptical and circular contact areas are provided. An example illustrates the influence of anisotropic effects on the evolution of dynamical characteristics of the elliptical and circular thin plates. Some remarks dedicated to the dynamic behavior of the mass point are presented.