The orthogonalized plane wave method is applied to the zincblende lattice. Crystal symmetrized combinations of plane waves are constructed for the zincblende lattice and the secular equations for the electronic levels at symmetry points of the reduced zone are derived as explicit functions of parameters depending on the crystal potential and on the core eigenstates. Calculations of valence and conduction eigenvalues and eigenfunctions are carried out on a number of group IV elements and of III-V compounds starting from the Hartree-Fock atomic core states and a model crystal potential constructed as a sum of atomic potentials in which the Slater approximation is used for the exchange contribution. The resulting band structures are very similar for all the semiconductors considered and the sequence of electronic levels confirms previous qualitative analyses. A comparison with experiments reveals, however, that those s-like conduction states which are most sensitive to the crystal potential are too high with respect to the other conduction states in the present approximation. An analysis of the approximation used indicates that the largest error in the calculations comes from the use of the Slater exchange in the model potential. Ways of improving on the accuracy of the calculations are suggested; one way is to use Hartree-Fock-Slater atomic results as a starting point. It is shown for the case of germanium that this improvement brings the results for the s-like conduction states into a closer agreement with experiment.