Propagating a short, relativistically intense laser pulse in a plasma channel makes it possible to generate clean comb-like electron beams – sequences of synchronized, low phase-space volume bunches with controllable energy spacing [S. Y. Kalmykov et al., “Accordion Effect Revisited: Generation of Comb-Like Electron Beams in Plasma Channels,” in Advanced Accelerator Concepts: 16th Workshop, AIP Conference Proceedings; this volume]. All-optical control of the electron beam phase space structure via manipulation of the drive pulse phase (negative chirp) and parameters of the channel enables the design of a tunable, all-optical source of polychromatic pulsed gamma-rays using the mechanism of inverse Compton scattering.