Laser cooling of heavy-ion beams in a storage ring is systematically studied with a multiparticle simulation code where not only exact lattice characteristics and space-charge forces but also realistic laser-ion interactions can be incorporated. The resonant coupling method is applied in order to extend the powerful longitudinal photon pressure to the transverse degrees of freedom. It is shown that, in spite of a space-charge-induced tune shift, the synchrobetatron resonance mechanism required for fast damping of transverse oscillations operates throughout the cooling process. Extremely efficient threedimensional cooling of stored ion beams is thus feasible. It is demonstrated that, at low line density, normalized root-mean-squared emittances of the order of 10-12 m·rad can be reached in all three directions by employing only existing technologies.