It is of key importance to study the detailed physics of AGN feedback, which plays a key role in the evolution of hot gas in galaxy clusters. Observational evidence for AGN feedback comes from a large number of quasi-spherical X-ray cavities and radio bubbles detected near cluster centers. Using numerical simulations following dynamical interactions of cosmic rays with thermal gas, we show that AGN jets producing these cavities are very light (e.g., jet-ICM density contrast ~ 0.0001), and energetically dominated by cosmic rays. More massive jets dominated by kinetic energy typically penetrate deep into the intracluster medium, forming cavities strongly elongated in the jet direction. We also show that cosmic ray diffusion across surfaces of radio bubbles should be significantly suppressed if the radio surface brightness declines sharply at the surfaces. This can be explained if magnetic fields near the surfaces are parallel to the surfaces and the cross-field diffusion is very small. Our calculations thus motivate future MHD simulations of anisotropic cosmic ray diffusion in galaxy clusters.