Based on the Hermite interpolation, a highly accurate method was proposed for the periodic motion of nonlinear conservative systems. It is shown that a Hermite interpolation solution for a dynamical system can be obtained by transforming the independent time variable to a vibration time. The corresponding transformed differential equation becomes well-conditioned for a solution by the Hermite interpolation method (HIM). The convergence and accuracy of the proposed HIM is superior to the traditional Qaisi’s power series method for Hermite interpolation using the information of two points instead of one point. By a way of illustration, the approximate general solutions of a class of nonlinear oscillators were derived by the HIM. The results show that the approximate general solutions can be used in the analysis on the vibration characteristics of oscillators with high accuracy.