Discontinuous reception (DRX), where a user equipment (UE) temporarily disables its receiver, is a critical power saving feature in modern cellular systems. DRX is likely to be particularly aggressively used in the mmWave and THz frequencies due to the high front end power consumption. A key challenge of DRX in these frequencies is that individual links are directional and highly susceptible to blockage. MmWave and THz UEs will therefore likely need to monitor multiple cells in multiple directions to ensure continuous reliable connectivity. This work proposes a novel, heuristic algorithm to dynamically select the cells to monitor to attempt to optimally trade-off link reliability and power consumption. The paper provides preliminary estimates of connected mode DRX mode consumption using detailed and realistic statistical models of blockers at both 28 and 140 GHz. It is found that although blockage dynamics are faster at 140 GHz, reliable connectivity at low power can be maintained with sufficient macro-diversity and link prediction.