High-redshift , dust-obscured galaxies – selected to be luminous in the radio but relatively faint at 850 \mu m – 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 \sim 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 ( 8.3 \times 10 ^ { 9 } M _ { \odot } , < 5.6 \times 10 ^ { 9 } M _ { \odot } and 15.4 \times 10 ^ { 9 } M _ { \odot } , respectively ) and , in the case of RG J163655 , a dynamical mass by measurement of the width of the CO ( 3–2 ) line ( 8 \times 10 ^ { 10 } \csc ^ { 2 } i M _ { \odot } ) . While these gas masses are substantial , on average they are 4 \times lower than submm-selected galaxies ( SMGs ) . Radio-inferred star formation rates ( < { SFR _ { radio } } > = 970 M _ { \odot } yr ^ { -1 } ) suggest much higher star-formation efficiencies than are found for SMGs , and shorter gas depletion time scales ( \sim 11 Myr ) , much shorter than the time required to form their current stellar masses ( \sim 160 Myr ; \sim 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 RG J163655 at 350 \mu m 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 .