We have presented a numerical technique for analyzing one-dimensional transient temperature distributions in a circular hollow cylinder that was composed of functionally graded ceramic-metal-based materials, without considering the temperature-dependent material properties. The functionally graded material (PGM) cylinder was assumed to be initially in a steady state of gradient temperature; the ceramic inner surface was exposed to high temperature, and the metallic outer surface, which was associated with its in-service performance, was exposed to low temperature. Then, the FGM cylinder was cooled rapidly on the ceramic surface of the cylinder, using a cold medium. The transient temperature and related thermal stresses in the FGM cylinder were analyzed numerically for a model of the mullite-molybdenum FGM system. The technique for analyzing the temperature distribution is quite simple and widely applicable for various boundary conditions of FGMs, in comparison with methods that have been proposed recently by other researchers.