  We identify and phase a sample of 107 Cepheids with 10<P/days<100 in M81
using the LBT and calibrate their BVI mean magnitudes with archival HST data.
The use of a ground-based telescope to identify and phase the Cepheids and HST
only for the final calibration reduces the demand on HST by nearly an order of
magnitude and yields Period-Luminosity (PL) relations with dispersions
comparable to the best LMC samples. We fit the sample using the OGLE-II LMC PL
relations and are unable to find a self-consistent distance for different band
combinations or radial locations within M81. We can do so after adding a radial
dependence to the PL zero point that corresponds to a luminosity dependence on
metallicity of g_mu=-0.56+/-0.36 mag/dex. We find marginal evidence for a shift
in color as a function of metallicity, distinguishable from the effects of
extinction, of g_2=+0.07+/-0.03 mag/dex. We find a distance modulus for M81,
relative to the LMC, of mu(M81-LMC)=9.39+/-0.14 mag, including uncertainties
due to the metallicity corrections. This corresponds to a distance to M81 of
3.6+/-0.2 Mpc, assuming a LMC distance modulus of 18.41 mag. We carry out a
joint analysis of M81 and NGC4258 Cepheids and simultaneously solve for the
distance of M81 relative to NGC4258 and the metallicity corrections. Given the
current data, the uncertainties of such joint fits are dominated by the
relative metallicities and the abundance gradients rather than by measurement
errors of the Cepheid magnitudes or colors. We find mu(M81-LMC)=9.40
(-0.11/+0.15) mag, mu(N4258-LMC)=11.08 (-0.17/+0.21) mag and
mu(N4258-M81)=1.68+/-0.08 mag and joint metallicity corrections of g_mu=-0.62
(-0.35/+0.31) mag/dex and g_2=0.01+/-0.01 mag/dex. Quantitative analyses of
Cepheid distances must take into account both the metallicity dependencies of
the Cepheids and the uncertainties in the abundance estimates. (ABRIDGED)
