A method of band mapping providing full control of the three-dimensional k is described in detail. Angle-dependent very-low-energy electron diffraction is applied to determine the photoemission final states along a Brillouin zone symmetry line parallel to the surface; photoemission out of these states is then utilized to map the valence bands in the constant-final-state mode. The method naturally incorporates the non-free-electron and excited-state self-energy effects in the unoccupied band, resulting in an accuracy superior over conventional techniques. Moreover, its intrinsic accuracy is less limited by lifetime broadening. As a practical advantage, the method provides access to a variety of lines in the Brillouin zone using only one crystal surface. We extensively tested the method on Cu. Several new aspects of the electronic structure of this metal are determined, including non-free-electron behavior of unoccupied bands and missing pieces of the valence band.