Results are presented from the development of a method for measuring plasma temperature in indirect (X-ray) drive targets by recording the shock wave velocity in the Iskra-5 facility. The samples under investigation were irradiated by X-rays in a converter box, and the shock wave velocity was determined from the time at which the wave reached the back surface of the sample and the surface began to emit visible radiation. This emission, in turn, was detected by a streak camera. The results of experiments on the interaction of X radiation with a hot dense plasma, as well as the accompanying gas-dynamic processes in aluminum samples, are analyzed both theoretically and numerically. In experiments with Al and Pb samples, the shock wave velocity was measured to vary in the range U = 8–35 km/s, and the range of variation of the temperature of the box walls was measured to be Te = 140–170 eV.