This paper introduces a fast Fourier transform (FFT)-based spectral dynamic analysis method for the transient responses as well as the steady-state responses of the linear discrete systems subject to non-zero initial conditions. The forced vibration of a viscously damped three-DOF system is considered as the illustrative numerical example. The proposed spectral analysis method is evaluated by comparing its results with the exact analytical solutions and the numerical solutions obtained by the Runge-Kutta method.