We report the density-functional-theory study of the electronic structure and exchange interactions in four diluted magnetic semiconductors (ZnMe)O, Me=Mn,Fe,Co,Ni. The calculations are performed for two impurity concentrations of 25 percent and 6.25 percent. For all systems the interatomic exchange interactions are short range with only the interaction between the nearest 3d impurities being sizable. For (ZnMn)O, the leading exchange interaction is antiferromagnetic for both impurity concentrations. In (ZnCo)O, the leading exhcange interaction is ferromagnetic for high concentration and antiferromagnetic for low concentration. (ZnFe)O and (ZnNi)O are ferromagnetic for both concentrations. In all cases of the antiferromagnetic exchange interaction, the energy bands are either completely filled or empty. This reveals the connection between the presence of the charge carriers and ferromagnetism.