We report on the time evolution of particle showers, as measured in modules of the uranium-scintillator barrel calorimeter of the ZEUS detector. The time development of hadronic showers differs significantly from that of electromagnetic showers, with about 40% of the response to hadronic showers arising from energy depositions which occur late in the shower development. The degree of compensation and the hadronic energy resolution were measured as a function of integration time, giving a value of e/pi = 1.02 +/- 0.01 for a gate width of 100 ns. The possibilities for electron-hadron separation based on the time structure of the shower were studied, with pion rejection factors in excess of 100 being achieved for electron efficiencies greater than 60%. The custom electronics used to perform these measurements samples the calorimeter signal at close to 60 MHz, stores all samples for a period of over 4 mus using analog switched capacitor pipelines, and digitizes the samples for triggered events with 12-bit ADCs.