The polar regions are predicted to experience the largest relative change in precipitation in response to increased greenhouse-gas concentrations, where a substantial absolute increase in precipitation coincides with small precipitation rates in the present-day climate. The reasons for this amplification, however, are still debated. Here, we use an atmospheric energy budget to decompose regional precipitation change from climate models under greenhouse-gas forcing into contributions from atmospheric radiative feedbacks, dry-static energy flux divergence changes, and surface sensible heat flux changes. The polar-amplified relative precipitation change is shown to be a consequence of the Planck feedback, which, when combined with larger polar warming, favors substantial atmospheric radiative cooling that balances increases in latent heat release from precipitation. Changes in the dry-static energy flux divergence contribute modestly to the polar-amplified pattern. Additional contributions to the polar-amplified response come, in the Arctic, from the cloud feedback and, in the Antarctic, from both the cloud and water vapor feedbacks. The primary contributor to the intermodel spread in the relative precipitation change in the polar region is also the Planck feedback, with the lapse rate feedback and dry-static energy flux divergence changes playing secondary roles. For all regions, there are strong covariances between radiative feedbacks and changes in the dry-static energy flux divergence that impact the intermodel spread. These results imply that constraining regional precipitation change, particularly in the polar regions, will require constraining not only individual feedbacks but also the covariances between radiative feedbacks and atmospheric energy transport.