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 (CO)-C-13( 2 -> 1), (CO)-C-12( 4 -> 3), and (CO)-C-12( 7 -> 6) with the Antarctic Submillimeter Telescope and Remote Observatory ( AST/RO). The cores, 10 of which contain embedded sources, are located mostly in the Vela, Ophiuchus, Lupus, Chamaeleon, Musca, and Scorpius complexes. (CO)-C-12( 7 -> 6) emission was undetected in all 40 clouds. We present data of 40 sources in (CO)-C-13( 2 -> 1) and (CO)-C-12( 4 -> 3), significant upper limits of (CO)-C-12( 7 -> 6), as well as a statistical analysis of the observed properties of the clouds. We find the typical (CO)-C-13( 2 -> 1) line width to be 2.0 km s(-1) for cores with embedded stars and 1.8 km s(-1) for all others. The typical 12CO( 4 -> 3) line width is 2.6-3.7 km s(-1) for cores with known embedded sources, and 1.6-2.3 km s(-1) for all others. The average (CO)-C-13 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 26K. 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 255M(circle dot) . Overall, our c2d sample of southern molecular cores has a range of properties ( line width, column density, size, mass, embedded stars) similar to those of past studies.