Observations suggest that a large fraction of black hole growth occurs in normal star-forming disk galaxies. Here we describe simulations of black hole accretion in isolated disk galaxies with sufficient resolution (~5 pc) to track the formation of giant molecular clouds that feed the black hole. Black holes in z=2 gas-rich disks (fgas=50%) occasionally undergo ~10 Myr episodes of Eddington-limited accretion driven by stochastic collisions with massive, dense clouds. We predict that these gas-rich disks host weak AGNs 1/4 of the time, and moderate/strong AGNs 10% of the time. Averaged over 100 Myr timescales and the full distribution of accretion rates, the black holes grow at a few per cent of the Eddington limit -- sufficient to match observations and keep the galaxies on the MBH-Mbulge relation. This suggests that dense cloud accretion in isolated z=2 disks could dominate cosmic black hole growth. In z=0 disks with fgas=10%, Eddington-limited growth is extremely rare because typical gas clouds are smaller and more susceptible to disruption by AGN feedback. This results in an average black hole growth rate in high-fgas galaxies that is up to 1000 times higher than that in low-fgas galaxies. In all our simulations, accretion shows variability by factors of 10^4 on a variety of time scales, with variability at 1 Myr scales driven by the structure of the interstellar medium.
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