Using the Australia Telescope Compact Array, we conducted a survey of CO J = 1 - 0 and J = 2 - 1 line emission towards strongly lensed high-redshift dusty star-forming galaxies (DSFGs) previously discovered with the South Pole Telescope (SPT). Our sample comprises 17 sources that had CO-based spectroscopic redshifts obtained with the Atacama Large Millimeter/submillimeter Array and the Atacama Pathfinder Experiment. We detect all sources with known redshifts in either CO J = 1 - 0 or J = 2 - 1. 12 sources are detected in the 7-mm continuum. The derived CO luminosities imply gas masses in the range (0.5-11) x 10(10) M-circle dot and gas depletion time-scales t(dep) < 200 Myr, using a CO to gas mass conversion factor alpha(CO) = 0.8 M-circle dot (K km s(-1) pc(2))(-1). Combining the CO luminosities and dust masses, along with a fixed gas-to-dust ratio, we derive alpha(CO) factors in the range 0.4-1.8 M-circle dot (K km s-1 pc2)-1, similar to what is found in other starbursting systems. We find small scatter in alpha(CO) values within the sample, even though inherent variations in the spatial distribution of dust and gas in individual cases could bias the dust-based alpha(CO) estimates. We find that lensing magnification factors based on the CO linewidth to luminosity relation (mu(CO)) are highly unreliable, but particularly when mu < 5. Finally, comparison of the gas and dynamical masses suggest that the average molecular gas fraction stays relatively constant at z = 2-5 in the SPT DSFG sample.