  Recent observations show that the cooling flows in the central regions of
galaxy clusters are highly suppressed. Observed AGN-induced cavities/bubbles
are a leading candidate for suppressing cooling, usually via some form of
mechanical heating. At the same time, observed X-ray cavities and synchrotron
emission point toward a significant non-thermal particle population. Previous
studies have focused on the dynamical effects of cosmic-ray pressure support,
but none have built successful models in which cosmic-ray heating is
significant. Here we investigate a new model of AGN heating, in which the
intracluster medium is efficiently heated by cosmic-rays, which are injected
into the ICM through diffusion or the shredding of the bubbles by
Rayleigh-Taylor or Kelvin-Helmholtz instabilities. We include thermal
conduction as well. Using numerical simulations, we show that the cooling
catastrophe is efficiently suppressed. The cluster quickly relaxes to a
quasi-equilibrium state with a highly reduced accretion rate and temperature
and density profiles which match observations. Unlike the conduction-only case,
no fine-tuning of the Spitzer conduction suppression factor f is needed. The
cosmic ray pressure, P_c/P_g <~ 0.1 and dP_c/dr <~ 0.1 \rho g, is well within
observational bounds. Cosmic ray heating is a very attractive alternative to
mechanical heating, and may become particularly compelling if GLAST detects the
gamma-ray signature of cosmic-rays in clusters.
