The magnetic interactions between organic pi -radicals adsorbed on 1D and 2D alpha -boron-nitride (alpha -BN) crystals are investigated theoretically. As in the case of analogous systems with graphite surfaces [N. Tyutyulkov, F. Dietz, K. Mullen, Chem. Phys. 255 (2000) 223], an intramolecular exchange interaction between the pi -electrons within a narrow half-filled band occurs, with the main component from the Coulomb (Hund) exchange. Compared to graphite the values of the effective Heisenberg exchange integral are smaller in the case of alpha -BN. The energy spectrum, and the magnetic and thermodynamic characteristics of Ising models consisting of methyl radicals adsorbed on 1D and 2D alpha -BN surfaces are discussed based on the analytically solved equations.