Forty molecular cloud cores in the southern hemisphere from the initial Spitzer Space Telescope Cores-to-Disks (c2d) Legacy program source list have been surveyed in 13CO(2-1), 12CO(4-3), and 12CO(7-6) with the Antarctic Submillimeter Telescope and Remote Observatory (AST/RO). The cores, ten of which contain embedded sources, are located mostly in the Vela, Ophiuchus, Lupus, Chamaeleon, Musca, and Scorpius complexes. 12CO(7-6) emission was undetected in all 40 clouds. We present data of 40

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... es in 13CO(2-1) and 12CO(4-3), significant upper limits of 12CO(7-6), as well as a statistical analysis of the observed properties of the clouds. We find the typical 13CO(2-1) linewidth to be 2.0 km/s for cores with embedded stars, and 1.8 km/s for all others. The typical 12CO(4-3) linewidth is 2.6 to 3.7 km/s for cores with known embedded sources, and 1.6 to 2.3 km/s for all others. The average 13CO column density derived from the line intensities was found to be 1.9 x 10^15 cm^(-2) for cores with embedded stars, and 1.5 x 10^15 cm^(-2) for all others. The average kinetic temperature in the molecular cores, determined through a Large Velocity Gradient analysis of a set of nine cores, has an average lower limit of 16 K and an average upper limit of 26 K. The average molecular hydrogen density has an average lower limit of 10^2.9 cm^(-3) and an average upper limit of 10^3.3 cm^(-3) for all cores. For a different subset of nine cores, we have derived masses. They range from 4 to 255 M_sun. Overall, our c2d sample of southern molecular cores has a range of properties (linewidth, column density, size, mass, embedded stars) similar to those of past studies.