The extended stochastic central difference (ESCD) method is proposed as a viable alternative for computing linear responses of discretized multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random disturbances. The method provides a means of controlling the center frequencies and bandwidths of narrow band stationary and nonstationary random excitation processes. It is suitable for larger-scale random response analysis of complicated structures idealized by the finite element method. Its additional important feature is that application of normal mode or complex normal mode analysis or direct numerical integration algorithms such as the fourth-order Runge-Kutta scheme is not required. Examples, including one of flow-induced vibration of a pipe containing a moving fluid are included to demonstrate: (1) the capability of the proposed method and difference between responses of discretized systems under narrow band and wide band random excitations, and (2) its accuracy and efficiency by way of comparison to the Monte Carlo simulation data. Generalization of the ESCD method for computation of responses of nonlinear mdof systems is presented in a companion paper.