Transition radiation is emitted when a perturbation source (e.g., electric charge, mechanical load), which does not possess an inherent frequency, moves along a straight line at a constant velocity in or near an inhomogeneous medium. The phenomenon was described for the first time in electromagnetics, but it is universal from the physical point of view. Transition radiation of elastic waves is emitted, for example, by a train running on a conventional railway track. The wheels of the train excite elastic waves in the track due to inhomogeneities such as non-uniform subsoil. Transition radiation of elastic waves has been studied in several 1D and 2D elastic systems, but the radiation in an elastic continuum has only been described in an idealized model consisting of two half-planes with constant load that crosses the interface. Nevertheless, the study provided physical insight into the mechanism of transition radiation. Body waves as well as interface waves can be excited, and the radiation spectra of the former show peculiar directivities due to the coupling of the waves at the interface. Here, we describe transition radiation in a more realistic continuum model of two elastic layers with a free surface. The constant load now moves along the free surface and passes over the interface of the two layers. The major difference from the above-mentioned model is the possible radiation of free-surface (Rayleigh) waves. In both layers, the radiation fields consist of a summation of guided modes, and the fields are coupled at the interface. Orthogonality relations derived from the elastodynamic reciprocity theorem are used to find the modal coefficients. Based on the derived solution, the spectra of radiation energy and their directivities can be calculated.