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This paper presents a new and efficient transient analysis method for mildly nonlinear circuits. The method is based on Volterra series representation of nonlinear circuits. It characterizes nonlinear circuits using a set of linear circuits called Volterra circuits. The input of the first-order Volterra circuit is identical to that of the nonlinear circuit, whereas that of higher order Volterra circuits is obtained from the response of lower order Volterra circuits. Fourier series interpolation is employed to approximate the input of higher order Volterra circuits. These circuits are analyzed using the sampled-data simulation of linear circuits for computational efficiency and the response of nonlinear circuits is obtained at equally spaced intervals of time. The accuracy of the method is controlled by the order of Volterra and interpolating Fourier series. Various sources contributing to the error are analyzed. The method has been implemented in a computer program. Numerical results on example circuits demonstrate that the accuracy of the method is comparable to that of linear multistep predictor-corrector algorithms, but with greatly improved speed.