Effective, solvent-mediated interaction potentials between alkali metal ions (Li+, Na+, K+, and Cs+) and selected sites on the DNA surface are derived from molecular dynamics simulations data. The effective potentials reproduce the corresponding ion-ion and ion-DNA radial distribution functions, obtained in the detailed full-atomistic molecular dynamics simulations. In this work, we have first calculated the ion-DNA and ion-ion radial distribution functions from molecular dynamics simulations of a periodic fragment of double-helix DNA in ionic aqueous solution. Thereafter we applied the inverse Monte Carlo method, previously suggested by us [Phys. Rev. E 52, 3730 (1995)], to derive the effective interaction potentials. Finally, a considerably larger DNA fragment was constructed together with its surrounding solvent environment and Monte Carlo simulations were performed to calculate the ion distributions and the relative binding affinities of different alkali ions to DNA. The results agree well with available experimental data.