In the paper, the unsteady dynamics of a thin elastic anisotropic unbounded Chow plate mounted on an elastic-inertial base has been studied. An anisotropy model where the elastic medium had one plane of symmetry geometrically coinciding with the median plane of the plate was considered. The base was specified by its stiffness coefficient and mass one. The problem statement included equations of motion in displacements, initial conditions and those of infinity. The solution of the problem was constructed using the method of fundamental solutions and integral Fourier transforms in spatial coordinates and Laplace ones in time. The Fourier originals were found numerically using the method of integrating rapidly oscillating functions. An example of the calculation was given.