The calculation of electric field gradient (EFG) lattice sums over point charges in ionic crystals is considered. Although the de Wette method gives rapidly converging sums under favourable circumstances, direct summations over a unit cell-shaped cavity are found to produce lattice sums which converge regularly as N-2 (N is the number of unit cells in a side of the cavity), allowing accurate extrapolation to their values for an infinite lattice by means of Neville tables. This convergence behaviour can be explained mathematically for orthogonal lattices using the Euler-Maclaurin formula. A point charge calculation of the EFGs at low symmetry sites in GdFe03 has been carried out to compare the convergence of the direct summation and de Wette techniques and to illustrate the N-2 convergence of the direct lattice sums.