Recently, a significant amount of transverse wave energy has been estimated propagating along solar atmospheric magnetic fields. However, these estimates have been made with the classic bulk Alfvén wave model which assumes a homogeneous plasma. In this paper, the kinetic, magnetic, and total energy densities and the flux of energy are computed for transverse MHD waves in one-dimensional cylindrical flux tube models with a piecewise constant or continuous radial density profile. There are fundamental deviations from the properties for classic bulk Alfvén waves. (1) There is no local equipartition between kinetic and magnetic energy. (2) The flux of energy and the velocity of energy transfer have, in addition to a component parallel to the magnetic field, components in the planes normal to the magnetic field. (3) The energy densities and the flux of energy vary spatially, contrary to the case of classic bulk Alfvén waves. This last property has the important consequence that the energy flux computed with the well known expression for bulk Alfvén waves could overestimate the real flux by a factor in the range 10-50, depending on the flux tube equilibrium properties.