Detections of luminous extended H-alpha emission in 14 distant X-ray-selected clusters of galaxies with redshifts ranging between 0.07 and 0.37 are reported. Eleven of these detections are from a complete flux-limited sample of 23 clusters extracted from the Einstein Extended Medium-Sensitivity Survey (EMSS). The H-alpha detections indicate the presence of cool gas embedded within hotter, X-ray-emitting cluster gas, a signature of a massive cooling flow. The X-ray and optical properties of these distant cooling flows are found to be similar to cooling flows found nearby (redshift value less than 0.1). If extended H-alpha emission is an unbiased indicator of a cooling flow, the fraction of X-ray-emitting clusters that possess massive cooling flows has decreased by a factor of about 2 since a redshift value of 0.3. The EMSS is rich in distant cooling flow clusters, not because of a selection effect as previously suggested but because cooling flow clusters comprised a large percentage of X-ray-emitting clusters in the past.