We present a five-band Herschel study (100-500 µm) of three galaxy clusters at z ∼ 1.2 from the Spitzer Adaptation of the Red-Sequence Cluster Survey (SpARCS). With a sample of 120 spectroscopically-confirmed cluster members, we investigate the role of environment on galaxy properties utilizing the projected cluster phase space (line-of-sight velocity versus clustercentric radius), which probes the time-averaged galaxy density to which a galaxy has been exposed. We divide cluster galaxies into phase-space bins of (r/r 200 ) × (∆v/σ v ), tracing a sequence of accretion histories in phase space. Stacking optically star-forming cluster members on the Herschel maps, we measure average infrared star formation rates, and, for the first time in high-redshift galaxy clusters, dust temperatures for dynamically distinct galaxy populations-namely, recent infalls and those that were accreted onto the cluster at an earlier epoch. Proceeding from the infalling to virialized (central) regions of phase space, we find a steady decrease in the specific star formation rate and increase in the stellar age of star-forming cluster galaxies. We perform a probability analysis to investigate all acceptable infrared spectral energy distributions within the full parameter space and measure a ∼ 4σ drop in the average dust temperature of cluster galaxies in an intermediate phase-space bin, compared to an otherwise flat trend with phase space. We suggest one plausible quenching mechanism which may be consistent with these trends, invoking ram-pressure stripping of the warmer dust for galaxies within this intermediate accretion phase.