A novel finite-difference time-domain (FDTD) approach is proposed for the analysis of wave propagation in a general frequency-dependent medium. In the proposed method, formulation of the fractional derivatives in the time-domain representation is circumvented by using the fast inverse Laplace transform (FILT) and Prony's method. The FILT is used to transform the dispersion expressed in a frequency-domain response into a time-domain response, and Prony's method is utilized to extract the parameters and transform the time-domain responses into those in the z-domain so that they can be incorporated into the FDTD method directly. The update equation of the electric field in the FDTD method is then formulated by using the z-transformation. Stability analysis of the proposed scheme is also investigated by means of the root-locus method. Reflection coefficients of dispersive media, such as Debye, Cole-Cole, Davidson-Cole, and Havriliak-Negami media simulated in a 1-D domain and a three-layered biological medium of skin, fat, and muscle tissues inside a waveguide with a TE10 fundamental mode in a 2-D domain, are found to be in good agreement with those obtained by an analytical method over a broad frequency range, demonstrating the validity of the proposed FDTD scheme.