[abridged] We present interferometric CO observations of twelve z~2 submillimetre-faint, star-forming radio galaxies (SFRGs) which are thought to be ultraluminous infrared galaxies (ULIRGs) possibly dominated by warmer dust (T_dust ~> 40 K) than submillimetre galaxies (SMGs) of similar luminosities. Four other CO-observed SFRGs are included from the literature, and all observations are taken at the Plateau de Bure Interferometer (PdBI) in the compact configuration. Ten of the sixteen SFRGs observed in CO (63%) are detected at >4sigma with a mean inferred molecular gas mass of ~2*10^10 M_sun. SFRGs trend slightly above the local ULIRG L_FIR-L_CO relation. Since SFRGs are about two times fainter in radio luminosity but exhibit similar CO luminosities to SMGs, this suggests SFRGs are slightly more efficient star formers than SMGs at the same redshifts. SFRGs also have a narrow mean CO line width, 320+-80km/s. SFRGs bridge the gap between properties of very luminous >5*10^12 L_sun SMGs and those of local ULIRGs and are consistent with intermediate stage major mergers. We suspect that more moderate-luminosity SMGs, not yet surveyed in CO, would show similar molecular gas properties to SFRGs. The AGN fraction of SFRGs is consistent with SMGs and is estimated to be 0.3+-0.1, suggesting that SFRGs are observed near the peak phase of star formation activity and not in a later, post-SMG enhanced AGN phase. This CO survey of SFRGs serves as a pilot project for the much more extensive survey of Herschel and SCUBA-2 selected sources which only partially overlap with SMGs. Better constraints on CO properties of a diverse high-z ULIRG population are needed from ALMA to determine the evolutionary origin of extreme starbursts, and what role ULIRGs serve in catalyzing the formation of massive stellar systems in the early Universe.
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