We present calculations of the photoionization (PI) cross sections of rubidium and cesium Rydberg atoms for light with wavelengths ranging from the infrared to the ultraviolet, using model potentials from [M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994)]. The origins of pronounced PI minima are identified by investigating the free-electron wavefunctions. These include broad PI minima in the nS to ǫP PI channels of both Rb and Cs, with free-electron energy ǫ, which are identified as Cooper minima. Much narrower PI minima in the nD to ǫF channels are due to shape resonances of the free-electron states. We describe possible experimental procedures for measuring the PI minima, and we discuss their implications in fundamental atomic physics as well as in practical applications.