In order to reduce computational effort and to allow for the use of periodic boundary conditions, electrostatic interactions in explicit solvent simulations of molecular systems do not obey Coulomb's law. Instead, a number of "effective potentials'' have been proposed, including truncated Coulomb, shifted, switched, reaction-field corrected, or Ewald potentials. The present study compares the performance of these schemes in the context of ionic solvation. To this purpose, a generalized form of the Born continuum model for ion solvation is developed, where ion-solvent and solvent-solvent interactions are determined by these effective potentials instead of Coulomb's law. An integral equation is formulated for calculating the polarization around a spherical ion from which the solvation free energy can be extracted. Comparison of the polarizations and free energies calculated for specific effective potentials and the exact Born result permits an assessment of the accuracy of these different schemes. Additionally, the present formalism can be used to develop corrections to the ionic solvation free energies calculated by molecular simulations implementing such effective potentials. Finally, an arbitrary effective potential is optimized to reproduce the Born polarization.