  We report the design, fabrication, and characterization of diamond
cantilevers attached to a phononic square lattice. We show that the robust
protection of mechanical modes by phononic band gaps leads to a
three-orders-of-magnitude increase in mechanical Q-factors, with the Q-factors
exceeding 10^6 at frequencies as high as 100 MHz. Temperature dependent studies
indicate that the Q-factors obtained at a few K are still limited by the
materials loss. The high-Q diamond nanomechanical resonators provide a
promising hybrid quantum system for spin-mechanics studies.
