Luminous, extended H-alpha emission has been detected in 14 distant X-ray-selected clusters of galaxies (0.07 < z < 0.37). Eleven of these detections are from a complete flux-limited sample (f(x) greater-than-or-equal-to 8 x 10(-13) ergs cm-2 s-1, delta greater-than-or-equal-to -20-degrees) 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. We draw several conclusions about the distant cooling flows revealed by their H-alpha emission. The X-ray and optical properties of these distant cooling flows are similar to cooling flows found nearby (z < 0.1). If extended H-alpha emission is an unbiased indicator of a cooling flow (i.e., the relationship between the presence of a cooling flow and detectable H-alpha emission does not change with redshift), then the fraction of X-ray-emitting clusters that possess massive cooling flows has decreased by a factor of about 2 since z approximately 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. There is limited evidence that the cosmological evolution of cooling flow clusters may be different from the non-cooling flow clusters. Also, accounting for the expected differences in the spatial extent of the X-ray emission of cooling flow and non-cooling flow clusters does not alter the conclusion of Gioia et al. that X-ray-luminous clusters have increased in number and/or luminosity since z approximately 0.3. One candidate luminous blue arc has been discovered in the course of this study.