  High-redshift, dust-obscured galaxies -- selected to be luminous in the radio
but relatively faint at 850um -- appear to represent a different population
from the ultra-luminous submillimeter- (submm-) bright population. They may be
star-forming galaxies with hotter dust temperatures or they may have lower
far-infrared luminosities and larger contributions from obscured active
galactic nuclei (AGN). Here we present observations of three z~2 examples of
this population, which we term submm-faint radio galaxies (SFRGs) in CO(3-2)
using the IRAM Plateau de Bure Interferometer to study their gas and dynamical
properties. We estimate the molecular gas mass in each of the three SFRGs
(8.3x10^{9} M_odot, <5.6x10^{9} M_odot and 15.4x10^{9} M_odot, respectively)
and, in the case of RG163655, a dynamical mass by measurement of the width of
the CO(3-2) line (8x10^{10} csc^2i M_odot). While these gas masses are
substantial, on average they are 4x lower than submm-selected galaxies (SMGs).
Radio-inferred star formation rates (<SFR_radio>=970 M_odot\yr) suggest much
higher star-formation efficiencies than are found for SMGs, and shorter gas
depletion time scales (~11 Myr), much shorter than the time required to form
their current stellar masses (~160 Myr; ~10^{11} M_odot). By contrast, SFRs may
be overestimated by factors of a few, bringing the efficiencies in line with
those typically measured for other ultraluminous star-forming galaxies and
suggesting SFRGs are more like ultraviolet- (UV-)selected star-forming galaxies
with enhanced radio emission. A tentative detection of \rga at 350um suggests
hotter dust temperatures -- and thus similar gas-to-dust mass fractions -- as
the SMGs. We conclude that SFRGs' radio luminosities are larger than would
naturally scale from local ULIRGs given their gas masses or gas fractions.
