Analog beamforming is an attractive and cost-effective solution to exploit the benefits of massive multiple-input-multiple-output systems, by requiring only one up/down-conversion chain. However, the presence of only one chain imposes a significant overhead in estimating the channel state information required for beamforming, when conventional digital channel estimation (CE) approaches are used. As an alternative, this paper proposes a novel CE technique, called periodic analog CE (PACE), that can be performed by analog hardware. By avoiding digital processing, the estimation overhead is significantly lowered and does not scale with number of antennas. PACE involves periodic transmission of a sinusoidal reference signal by the transmitter, estimation of its amplitude and phase at each receive antenna via analog hardware, and using these estimates for beamforming. To enable such non-trivial operation, two reference tone recovery techniques and a novel receiver architecture for PACE are proposed and analyzed, both theoretically and via simulations. Results suggest that in sparse, wide-band channels and above a certain signal-to-noise ratio, PACE aided beamforming suffers only a small loss in beamforming gain and enjoys a much lower CE overhead, in comparison to conventional approaches. Benefits of using PACE aided beamforming during the initial access phase are also discussed.