We analyze space-based time-series photometry of Sun-like stars, mostly in the Pleiades, but also field stars and the Sun itself. We focus on timescales between roughly 1 hr and 1 day. In the corresponding frequency band these stars display brightness fluctuations with a decreasing power-law continuous spectrum. K2 and Kepler observations show that the rms flicker due to this mid-frequency continuum (MFC) can reach almost 1%, approaching the modulation amplitude from active regions. The MFC amplitude varies by a factor up to 40 among Pleiades members with similar Teff, depending mainly on the stellar Rossby number Ro. For Ro ≤ 0.04, the mean amplitude is roughly constant at about 0.4%; at larger Ro the amplitude decreases rapidly, shrinking by about two orders of magnitude for Ro ≃ 1. Among stars, the MFC amplitude correlates poorly with that of modulation from rotating active regions. Among field stars observed for 3 yr by Kepler, the quarterly average modulation amplitudes from active regions are much more time variable than the quarterly MFC amplitudes. We argue that the process causing the MFC is largely magnetic in nature and that its power-law spectrum comes from magnetic processes distinct from the star's global dynamo, with shorter timescales. By analogy with solar phenomena, we hypothesize that the MFC arises from a (sometimes energetic) variant of the solar magnetic network, perhaps combined with rotation-related changes in the morphology of supergranules.