We present 12CO J = 3→2 maps of NGC 2976 and NGC 3351 obtained with the James Clerk Maxwell Telescope (JCMT), both early targets of the JCMT Nearby Galaxy Legacy Survey (NGLS). We combine the present observations with 12CO J = 1→0 data and find that the computed 12CO J = 3→2 to 12CO J = 1→0 line ratio (R31) agrees with values measured in other NGLS field galaxies. We compute the MH2 value and find that it is robust against the value of R31 used. Using H I data from The H I Nearby Galaxy Survey, we find a tight correlation between the surface density of H2 and star formation rate density for NGC 3351 when 12CO J = 3→2 data are used. Finally, we compare the 12CO J = 3→2 intensity with the polycyclic aromatic hydrocarbon (PAH) 8 μm surface brightness and find a good correlation in the high surface brightness regions. We extend this study to include all 25 Spitzer Infrared Nearby Galaxies Survey galaxies within the NGLS sample and find a tight correlation at large spatial scales. We suggest that both PAH 8 μm and 12CO J = 3→2 are likely to originate in regions of active star formation.