We present new observations of ¹²CO (J = 2-1), ¹³CO (J = 2-1), and 1.3 mm dust continuum emission toward seven diverse galaxies (IC 10, IRAS 10565+2448, the Antennae—NGC 4038 and 4039, NGC 4736, Mrk 231, and NGC 6090). Compared with the other galaxies in the sample, the two ultraluminous infrared galaxies (ULIGs) exhibit reduced ¹³CO line luminosities with respect to the ¹²CO line luminosities and 1.3 mm dust luminosities. These observations parallel the established trend of large R-values in ULIGs compared with more quiescent spiral galaxies. The two luminous merging pairs in our sample with moderate starburst activity, the Antennae and NGC 6090, do not appear to show the large R-values. The results are consistent with models in which ¹²CO emission is dominated in the most luminous mergers by warm regions of ISM with moderate optical depths (τ ~ 1). For the other galaxies, R is more typical of the optically thick case 10 < R < 20, and most of the emission probably arises from within cooler molecular clouds. We compare these data with previous ¹²CO (J = 3-2) and [C I] (³P₁-³P₀) observations from the literature.