  We present a first complete 12 CO J=3-2 map of M81, observed as part of the
Nearby Galaxies Legacy Survey being carried out at the James Clerk Maxwell
Telescope. We detect 9 regions of significant CO emission located at different
positions within the spiral arms, and confirm that the global CO emission in
the galaxy is low. We combine these data with a new H-alpha map obtained using
the Isaac Newton Telescope and archival HI, 24 microns and FUV images to
uncover a correlation between the molecular gas and star forming regions in
M81. For the nine regions detected in CO J=3-2, we combine our CO J=3-2 data
with existing CO J=1-0 data to calculate line ratios. We find that the ratio
J=(3-2)/(1-0) is in agreement with the range of typical values found in the
literature (0.2-0.8). Making reasonable assumptions, this allows us to
constrain the hydrogen density to the range (10^3-10^4) cm^{-3}. We also
estimated the amount of hydrogen produced in photo-dissociation regions near
the locations where CO J=3-2 was detected.
