We developed a method for calculating the ground-state properties and fundamental band-gaps of solids, using a generalized Kohn-Sham approach combining a local density approximation (LDA) functional with a long-range explicit exchange orbital functional. We found that when the range parameter is selected according to the formula γ = A/(ɛ∞-) where ɛ∞ is the optical dielectric constant of the solid and = 0.84 and A = 0.216 a0-1, predictions of the fundamental band-gap close to the experimental values are obtained for a variety of solids of different types. For most solids the range parameter γ is small (i.e. explicit exchange is needed only at long distances) so the predicted values for lattice constants and bulk moduli are similar to those based on conventional LDA calculations. Preliminary calculations on silicon give a general band structure in good agreement with experiment.