We demonstrate two-dimensional photonic crystal cavities operating at telecommunication wavelengths in a single-crystal diamond membrane. We use a high-optical-quality and thin (~ 300 nm) diamond membrane, supported by a polycrystalline diamond frame, to realize fully suspended two-dimensional photonic crystal cavities with a high theoretical quality factor of ~ $8times10^6$ and a relatively small mode volume of ~2$({lambda}/n)^3$. The cavities are fabricated in the membrane using electron-beam lithography and vertical dry etching. We observe cavity resonances over a wide wavelength range spanning the telecommunication O- and S-bands (1360 nm-1470 nm) with Q factors of up to ~1800. Our method offers a new direction for on-chip diamond nanophotonic applications in the telecommunication-wavelength range.