In this paper we present results from an IRAM Plateau de Bure millimetre-wave Interferometer (PdBI) survey for CO emission towards radio-detected submillimetre galaxies (SMGs) with known optical and near-infrared spectroscopic redshifts. Five sources in the redshift range z~1-3.5 were detected, nearly doubling the number of SMGs detected in CO. We summarise the properties of all 12 CO-detected SMGs, as well as 6 sources not detected in CO by our survey, and use this sample to explore the bulk physical properties of the SMG population as a whole. The median CO line luminosity of the SMGs is <L_CO> = (3.8 +- 2.0) x 10^10 K km/s pc^2. Using a CO-to-H_2 conversion factor appropriate for starburst galaxies, this corresponds to a molecular gas mass <M(H_2)> = (3.0 +- 1.6) x 10^10 Msun within a ~2kpc radius, about four times greater than the most luminous local ultraluminous infrared galaxies (ULIRGs) but comparable to that of the most extreme high-redshift radio galaxies and QSOs. The median CO fwhm linewidth is broad, <fwhm> = 780 +- 320 km/s, and the SMGs often have double peaked line profiles, indicative of either a merger or a disk. From their median gas reservoirs (~3 x 10^10 Msun) and star-formation rates (>700 Msun/yr) we estimate a lower limit on the typical gas-depletion time scale of >40Myr in SMGs. This is marginally below the typical age expected for the starbursts in SMGs, and suggests that negative feedback processes may play an important role in prolonging the gas consumption time scale. We find a statistically-significant correlation between the far-infrared and CO luminosities of the SMGs which extends the observed correlation for local ULIRGs to higher luminosities and higher redshifts. [ABRIDGED]
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