Hot, underdense bubbles powered by active galactic nuclei (AGN) are likely to play a key role in halting catastrophic cooling in the centers of cool-core galaxy clusters. We present three-dimensional simulations that capture the evolution of such bubbles, using an adaptive-mesh hydrodynamic code, FLASH3, to which we have added a subgrid model of turbulence and mixing. Pure-hydro simulations indicate that AGN bubbles are disrupted into resolution-dependent pockets of underdense gas. However, proper modeling of subgrid turbulence shows that Rayleigh-Taylor instabilities act to mix the heated regions with their surroundings, while at the same time preserving them as coherent structures, consistent with observations. Thus bubbles are transformed into hot clouds of mixed material as they move outwards in the hydrostatic intracluster medium. Properly capturing the evolution of such clouds has important implications for many ICM properties.