  We have obtained multi-epoch observations of 31 O-type stars in the Carina
Nebula using the CHIRON spectrograph on the CTIO/SMARTS 1.5-m telescope. We
measure their radial velocities to 1--2 km s$^{-1}$ precision and present new
or updated orbital solutions for the binary systems HD 92607, HD 93576, HDE
303312, and HDE 305536. We also compile radial velocities from the literature
for 32 additional O-type and evolved massive stars in the region. The combined
data set shows a mean heliocentric radial velocity of 0.6 km s$^{-1}$. We
calculate a velocity dispersion of $\le9.1$ km s$^{-1}$, consistent with an
unbound, substructured OB association. The Tr 14 cluster shows a marginally
significant 5 km s$^{-1}$ radial velocity offset from its neighbor Tr 16, but
there are otherwise no correlations between stellar position and velocity. The
O-type stars in Cr 228 and the South Pillars region have a lower velocity
dispersion than the region as a whole, supporting a model of distributed
massive-star formation rather than migration from the central clusters. We
compare our stellar velocities to the Carina Nebula's molecular gas and find
that Tr 14 shows a close kinematic association with the Northern Cloud. In
contrast, Tr 16 has accelerated the Southern Cloud by 10--15 km s$^{-1}$,
possibly triggering further massive-star formation. The expansion of the
surrounding H II region is not symmetric about the O-type stars in radial
velocity space, indicating that the ionized gas is constrained by denser
material on the far side.
