With its host of outstanding material properties, single-crystal diamond is an attractive material for nanomechanical systems. Here, the mechanical resonance characteristics of freestanding, single-crystal diamond nanobeams fabricated by an angled-etching methodology are reported. Resonance frequencies displayed evidence of significant compressive stress in doubly clamped diamond nanobeams, while cantilever resonance modes followed the expected inverse-length-squared trend. Q-factors on the order of 104 were recorded in high vacuum. Results presented here represent initial groundwork for future diamond-based nanomechanical systems which may be applied in both classical and quantum applications.