The evaluation of thermally induced stress wave propagation in solids and materials influenced by non-Fourier effects is described. Several pathological anomalies exist for the classical dynamic thermoelastic models using the Fourier heat conduction model, especially for cases involving extremely short transients or for temperatures near absolute zero. As a consequence, various modified theories have been proposed to account for finite speeds of thermal stress wave propagation, in contrast to investigations based on the classical thermoelastic theory, which allows thermal disturbances to propagate only at infinite speeds. The dynamic thermoelastic models used herein can be obtained from those of Green and Lindsay, which permit the so-called 'second sound' effects by appropriate choice of relaxation parameters. The fundamental purpose of this study is to provide accurate solutions to a class of thermally induced non-Fourier models in dynamic thermoelasticity which can help us to understand the representative behavior of the nature and mechanisms of the resulting thermal stress wave disturbances. In this regard, the present paper uses specially tailored hybrid formulations based on the transfinite element approach for accurately modeling the discontinuous thermal stress wave disturbances. The classical Fourier models of dynamic thermoelasticity which can be obtained by appropriate choice of relaxation parameters are also presented. The results obtained indicate that significant thermal stresses may arise because of non-Fourier effects, especially when the speeds of propagation of the thermal and stress waves are equal. For the case of unequal speeds of propagation, the relative magnitudes of the resulting thermal stress waves seem comparable to a certain extent with those obtained from the classical theory of dynamic thermoelasticity.