A solvable three-dimensional model is developed for band-structure calculations in valence crystals. The local atomic potential at each lattice site is replaced by a simpler potential operator which in isolation gives a finite number of bound states. Furthermore, an algorithm is presented for constructing this operator such that the bound-state wave functions and energies exactly match those of any given local atomic potential. A sample calculation is carried out for a "Gaussian atom" set in a fcc lattice where the two parameters, range and strength, have been chosen to fit the atomic radius and the first ionization potential of a Ge atom.