We demonstrate Kerr-frequency-comb generation with nanofabricated Fabry-Perot resonators, which are formed with photonic-crystal-reflector (PCR) mirrors. The PCR group-velocity dispersion (GVD) is engineered to counteract the strong normal GVD of a rectangular waveguide, fabricated on a thin, 450 nm silicon nitride device layer. The reflectors enable resonators with both high optical quality factor and anomalous GVD, which are required for Kerr-comb generation. We report design, fabrication, and characterization of devices in the 1550 nm wavelength bands. Kerr-comb generation is achieved by exciting the devices with a continuous-wave laser. The versatility of PCRs enables a general design principle and a material-independent device infrastructure for Kerr-nonlinear-resonator processes, opening new possibilities for manipulation of light. Visible and multispectral-band resonators appear to be natural extensions of the PCR approach.