This paper presents a numerical simulation of guided ultrasonic waves propagating in aluminum-epoxy-aluminum adhesive plates. In particular, this study investigated the effect of the epoxy thickness on the dispersive patterns, such as the phase velocity and group velocity of guided ultrasonic waves. In addition to investigating the dispersive curves, a numerical simulation using the pulse-echo method was carried out. This simulation showed that the degree of sensitivity of the epoxy thickness is dependent on the curvature of the phase and group dispersion curves, the maximum amplitude of the received time signals, and the peak frequency of the real components of the Fourier transform. Then, the linear relations between the epoxy thickness and the received and transformed signals were constructed to estimate the epoxy thickness.