In this paper, linearized approximations of both the forward and the inverse problems of resonant ultrasound spectroscopy for the determination of mechanical properties of thin surface layers are presented. The linear relations between the frequency shifts induced by the deposition of the layer and the in-plane elastic coefficients of the layer are derived and inverted, the applicability range of the obtained linear model is discussed by a comparison with nonlinear models and finite element method (FEM), and an algorithm for the estimation of experimental errors in the inversely determined elastic coefficients is described. In the final part of the paper, the linearized inverse procedure is applied to evaluate elastic coefficients of a 310 nm thick diamond-like carbon layer deposited on a silicon substrate.