We present an analysis of the levels and evolution of star formation activity in a representative sample of 30 massive galaxy clusters at 0.1510^10 M_sun) star-forming cluster galaxies within r200 are found to be systematically 28% lower than their counterparts in the field at fixed stellar mass and redshift, a difference significant at the 8.7-sigma level. This is the unambiguous signature of star formation in most (and possibly all) massive star-forming galaxies being slowly quenched upon accretion into massive clusters, their SFRs declining exponentially on quenching time-scales in the range 0.7-2.0 Gyr. We measure the mid-infrared Butcher-Oemler effect over the redshift range 0.0-0.4, finding rapid evolution in the fraction (f_SF) of massive (M_K3M_sun/yr, of the form f_SF (1+z)^7.6. We dissect the origins of the Butcher-Oemler effect, revealing it to be due to the combination of a ~3x decline in the mean specific-SFRs of star-forming cluster galaxies since z~0.3 with a ~1.5x decrease in number density. Two-thirds of this reduction in the specific-SFRs of star-forming cluster galaxies is due to the steady cosmic decline in the specific-SFRs among those field galaxies accreted into the clusters. The remaining one-third reflects an accelerated decline in the star formation activity of galaxies within clusters. The slow quenching of star-formation in cluster galaxies is consistent with a gradual shut down of star formation in infalling spiral galaxies as they interact with the intra-cluster medium via ram-pressure stripping or starvation mechanisms. We find no evidence for the build-up of cluster S0 bulges via major nuclear star-burst episodes. ; Comment: 24 pages, 12 figures. Accepted for publication in ApJ