The temperature-dependent behavior of excitonic photoluminescence observed in ZnO/MgZnO multiple quantum wells (MQWs) in the temperature range of 5-300 K is described. In a ZnO/Mg0.27Zn0.73O MQW grown by laser molecular-beam epitaxy, the luminescence was dominated by localized exciton (LE) emission throughout the whole temperature range studied. Luminescence of free excitons (FEs) was not observed. A simple rate equation is used to describe the quenching of LE emission. The activation energy for LE luminescence quenching is of the order of the localization energy of excitons, suggesting that the thermionic emission of the LEs out of the localization potentials leads to nonradiative recombination. In a ZnO/Mg0.12Zn0.88O MQW having lower barriers, the luminescence was dominated by LE emissions at low temperatures, while the FE transition was dominating emissions at temperatures above 175 K. A rate equation assuming one nonradiative recombination channel is used to describe the quenching of the transitions observed. The activation energy for LE luminescence quenching deduced in this sample is also of the order of the localization energy of excitons. The temperature dependences of FE emission intensities are also discussed by using a simple rate equation in which a thermal release effect of LEs toward FEs is taken into account.