Raman lasers based on bulk diamond are a valuable resource for generating coherent light in wavelength regimes where no common laser diodes are available. Nevertheless, the widespread use of such lasers is limited by their high threshold power requirements on the order of several Watts. Using on-chip microresonators, a significant reduction of the lasing threshold by more than two orders of magnitude has been shown. However, these resonators lack a continuous tuning mechanism and, mainly due to fabrication limitations, their implementation in the visible remains elusive. Here, we propose a platform for a diamond Raman laser in the visible. The device is based on a miniaturized, open-access Fabry-Perot cavity. Our microcavity provides widely-tunable doubly-resonant enhancement of Raman scattering from high quality single-crystalline diamond. We demonstrate a $>$THz continuous tuning range of doubly-resonant Raman scattering, a range limited only by the reflective stopband of the mirrors. Based on the experimentally determined quality factors exceeding $300\,000$, our theoretical analysis suggests that, with realistic improvements, a sub-mW threshold is readily within reach. Our findings pave the way to the creation of a universal low-power frequency shifter, a potentially valuable addition to the nonlinear optics toolbox.