Massive multi-antenna millimeter wave (mmWave) and terahertz wireless systems promise high-bandwidth communication to multiple user equipment in the same time-frequency resource. The high path loss of wave propagation at such frequencies and the fine-grained nature of beamforming with massive antenna arrays necessitates accurate channel estimation to fully exploit the advantages of such systems. In this paper, we propose BEAmspace CHannel EStimation (BEACHES), a low-complexity channel estimation algorithm for multi-antenna mmWave systems and beyond. BEACHES leverages the fact that wave propagation at high frequencies is directional, which enables us to denoise the (approximately) sparse channel state information in the beamspace domain. To avoid tedious parameter selection, BEACHES includes a computationally-efficient tuning stage that provably minimizes the mean-square error of the channel estimate in the large-antenna limit. To demonstrate the efficacy of BEACHES, we provide simulation results for line-of-sight (LoS) and non-LoS mmWave channel models.