This paper introduces a new numerical hybrid method to simulate transient wave propagation in a multilayer semi-infinite medium, which can be fluid or solid, subjected to given transient loads. The medium is constituted of a finite number of unbounded layers with finite thicknesses. The method has a low numerical cost and is relatively straightforward to implement, as opposed to most available numerical techniques devoted to similar problems. The proposed method is based on a time-domain formulation associated with a 2D-space Fourier transform for the variables associated with the two infinite dimensions and uses a finite element approximation in the direction perpendicular to the layers. An illustration of the method is given for an elasto-acoustic wave propagation problem: a three-layer medium constituted of an elastic layer sandwiched between two acoustic fluid layers and excited by an acoustic line source located in one fluid layer.