Abstract.We have mapped the CO(3-2) line emission from several nearby galaxies, using the Heinrich-HertzTelescope on Mt. Graham, Arizona. Unlike earlier observations, our investigation is not restricted to starburst galaxies, but includes twelve galaxies of various types and in different stages of star forming activity. Furthermore, we have not only observed the central positions of these objects, but have obtained maps of the extended CO(3-2) emission, with a typical map extent of 2 to 3 arcmin in each direction. Our observations show that this extended mapping is necessary to reveal spatial changes of the ISM properties within the galaxies. In this paper we present the data sets and some data analysis. We compare the galaxies in view of their morphology and excitation conditions, using line ratios, luminosities and other properties, like the extent of the CO(3-2) emission. The main results of this CO(3-2) survey are: 1. In none of the observed objects the emission is confined to the nucleus, as claimed in some earlier publications. CO(3-2) emission can be detected for some objects to the same extent as the CO(2-1) and the CO(1-0) lines. 2. The emission is more concentrated to the vicinity of star forming structures (nuclear regions and spiral arms) than the lower CO transitions for most of the observed objects. This is shown by decreasing (3-2)/(1-0) line intensity ratios from the very centres towards larger radii. The (deconvolved) sizes of the central emission peaks in the CO(3-2) line vary from about 300 pc up to 3 kpc. 3. The CO(3-2) luminosity is stronger in objects that contain a nuclear starburst or morphological peculiarities. The total power emitted in the CO(3-2) line from the central regions (i.e. excluding spiral arms/outer disk) is highest in the starburst galaxies NGC 2146, M 82, NGC 3628, and in the spiral galaxy M 51. When comparing the total power normalized to the size of the emission region, the starburst galaxies M 82 and NGC 253 show the highest values (about three times higher than most other objects), while NGC 278 and NGC 4631 show the lowest. 4. With the present spatial resolution, the line ratios R3,1 seem to be independent of Hubble type, color or luminosity. Most galaxies with enhanced central star formation ("starbursts") show line ratios of the integrated intensities of R3,1 ∼ 1.3 in the very centre and ∼1.0 at a radius of about 1 kpc. Objects with a ring-like (or double-peak if seen edge-on) molecular gas distribution (NGC 253, M 82, and NGC 4631) show lower ratios. The two galaxies that have CO(3-2) emission distributed over their spiral arms (NGC 891 and M 51) show very low line ratios despite their high infrared luminosities. This result suggests that CO emission in these objects reflects a large amount of molecular gas, but not enhanced star forming activity. 5. Starburst galaxies show CO(3-2) emission also in their disks. The line intensities are higher than that of normal galaxies. This suggests that even if a starburst is a localized phenomenon, it is related to different...