This paper deals with three-dimensional gravity driven free surface flows of piles of granular materials along bottom profiles that are weakly curved downward and plane laterally. We present in detail a three-dimensional extension of the two-dimensional Savage-Hutter model for such granular avalanches. In this extended model, the avalanche is described as a three-dimensional incompressible continuum obeying a Coulomb dry friction law at the base and a Mohr-Coulomb plastic yield criterion in the interior. Based on this, the balance laws of mass and linear momentum and kinematic and stress boundary conditions at the free surface and the base are used to derive depth-averaged dynamic equations that describe the temporal evolution of the height and the depth-averaged horizontal velocity components as functions of position and time. A computation is performed for a pile of granular material with an initial spherical cap geometry moving down an inclined plane.