(abridged) We have obtained precise radial velocities for a sample of 373 G and K type giants at Lick Observatory regularly over more than 12 years. Planets have been identified around 15 giant stars; an additional 20 giant stars host planet candidates. We investigate the occurrence rate of substellar companions around giant stars as a function of stellar mass and metallicity. We probe the stellar mass range from about 1 to beyond 3 M_Sun, which is not being explored by main-sequence samples. We fit the giant planet occurrence rate as a function of stellar mass and metallicity with a Gaussian and an exponential distribution, respectively. We find strong evidence for a planet-metallicity correlation among the secure planet hosts of our giant star sample, in agreement with the one for main-sequence stars. However, the planet-metallicity correlation is absent for our sample of planet candidates, raising the suspicion that a good fraction of them might indeed not be planets. Consistent with the results obtained by Johnson for subgiants, the giant planet occurrence rate increases in the stellar mass interval from 1 to 1.9 M_Sun. However, there is a maximum at a stellar mass of 1.9 +0.1/-0.5 M_Sun, and the occurrence rate drops rapidly for masses larger than 2.5-3.0 M_Sun. We do not find any planets around stars more massive than 2.7 M_Sun, although there are 113 stars with masses between 2.7 and 5 M_Sun in our sample (corresponding to a giant planet occurrence rate < 1.6% at 68.3% confidence in that stellar mass bin). We also show that this result is not a selection effect related to the planet detectability being a function of the stellar mass. We conclude that giant planet formation or inward migration is suppressed around higher mass stars, possibly because of faster disk depletion coupled with a longer migration timescale.
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