  We present the results of a spectroscopic survey of the kinematic structure
of star-forming galaxies at redshift z ~ 2 - 3 using Keck/OSIRIS integral field
spectroscopy. Our sample is comprised of 12 galaxies between redshifts z ~ 2.0
and 2.5 and one galaxy at z ~ 3.3 which are well detected in either HAlpha or
[O III] emission. These observations were obtained in conjunction with the Keck
laser guide star adaptive optics system, with a typical angular resolution
after spatial smoothing ~ 0.15" (approximately 1 kpc at the redshift of the
target sample). At most five of these 13 galaxies have spatially resolved
velocity gradients consistent with rotation while the remaining galaxies have
relatively featureless or irregular velocity fields. All of our galaxies show
local velocity dispersions ~ 60 - 100 km/s, suggesting that (particularly for
those galaxies with featureless velocity fields) rotation about a preferred
axis may not be the dominant mechanism of physical support. While some galaxies
show evidence for major mergers such evidence is unrelated to the kinematics of
individual components (one of our strongest merger candidates also exhibits
unambiguous rotational structure), refuting a simple bimodal disk/merger
classification scheme. We discuss these data in light of complementary surveys
and extant UV-IR spectroscopy and photometry, concluding that the dynamical
importance of cold gas may be the primary factor governing the observed
kinematics of z ~ 2 galaxies. We conclude by speculating on the importance of
mechanisms for accreting low angular-momentum gas and the early formation of
quasi-spheroidal systems in the young universe.(abridged)
