We present the design, fabrication, and characterization of high quality factor and small mode volume planar photonic crystal cavities from cubic (3C) thin films (thickness ~ 200 nm) of silicon carbide (SiC) grown epitaxially on a silicon substrate. We demonstrate cavity resonances across the telecommunications band, with wavelengths from 1,250 - 1,600 nm. Finally, we discuss possible applications in nonlinear optics, optical interconnects, and quantum information science.