We design, fabricate and study a novel platform for cavity optomechanics: a silicon nitride membrane patterned as a sub-wavelength diffraction grating. Using the grating as one mirror of a Fabry-Perot cavity, we realize an optical finesse of F = 2830 +/- 60, corresponding to a grating reflectivity of R = 0.998. The finesse we achieve appears to be within a factor of two of the limit set by material absorption. We study the finesse as a function of wavelength and optical spot size in order to elucidate various optical loss mechanisms. We find that the cavity exhibits birefringence, and establish that it, too, is a source of optical loss. We then characterize the mechanical motion. We observe hundreds of normal modes, and find the fluctuating amplitude of one of them to be very well described by a Boltzmann distribution. By injecting a red-detuned cooling laser, we optically cool all of the modes that we observe. The lowest effective temperature we achieve is T-eff approximate to 1 K.