According to our original discovery, the oscillatory course of the Cu2+-catalyzed oxidation of thiocyanate ions with hydrogen peroxide, in nonstirred medium and upon the addition of luminol as an indicator, can be a source of a novel type of dissipative patterns - luminescent traveling waves. The formation of these fronts, contrary to the patterns associated with the Belousov-Zhabotinsky reaction, cannot be explained in terms of coupled homogeneous kinetics and diffusion, and under isothermal conditions. Both experimental studies and numerical simulations of the kinetic mechanism suggest that the spatial progress of these waves requires mainly the temperature gradient in the solution, which affects the local chemical reaction rate (and thus the oscillation period), with practically negligible contribution from diffusion of reagents. As a consequence of this thermokinetic coupling, the observed traveling patterns are thus essentially the phase (or kinematic) waves, formed due to the spatial phase shift of the oscillations caused by differences in chemical reaction rates. The temperature gradient, caused by the significant heat effect of exothermic oxidation of thiocyanate by hydrogen peroxide, can emerge spontaneously as a local fluctuation or can be forced externally, if the control of progress of the achieved. luminescent waves is to be achieved.