Si3N4 is an excellent material for applications of nanophotonics at visible wavelengths due to its wide bandgap and moderately large refractive index (n $approx$ 2.0). We present the fabrication and characterization of Si3N4 photonic crystal nanobeam cavities for coupling to diamond nanocrystals and Nitrogen-Vacancy centers in a cavity QED system. Confocal micro-photoluminescence analysis of the nanobeam cavities demonstrates quality factors up to Q ~ 55,000, which is limited by the resolution of our spectrometer. We also demonstrate coarse tuning of cavity resonances across the 600-700nm range by lithographically scaling the size of fabricated devices. This is an order of magnitude improvement over previous SiNx cavities at this important wavelength range.