We show that the photonic analogue of the Korringa-Kohn-Rostoker method is a viable alternative to the plane-wave method for analysing the spectrum of electromagnetic waves in a three-dimensional periodic dielectric lattice. Firstly, in the case of a fcc lattice of homogeneous dielectric spheres, we reproduce the main feature of the spectrum obtained by the plane-wave method, namely that for a sufficiently high dielectric contrast a full gap opens in the spectrum between the eighth and ninth bands if the dielectric constant es of the spheres is lower than the dielectric constant eb of the background medium. If es>eb, no gap is found in the spectrum. The maximal value of the relative band-gap width approaches 14% in the close-packed case and decreases monotonically as the filling fraction decreases. The lowest dielectric contrast eb/es for which a full gap opens in the spectrum is determined to be 8.13. Eventually, in the case of a fcc lattice of coated spheres, we demonstrate that a suitable coating can enhance gap widths by as much as 50%.