We report the parameter-free, density-functional theory calculations of the electronic structure, interatomic exchange interaction and magnetic critical temperature of Zn_0.75Cr_0.25Te. This system has recently gained exceptional importance as the first diluted magnetic semiconductor (DMS), where the intrinsic ferromagnetism with Curie temperature higher than room temperature is confirmed by magnetic circular dichroism measurements. We obtain a value for the Curie temperature that is in good agreement with the experiment. The role of the holes in the valence band in mediating the ferromagnetic exchange interactions is demonstrated. The application of the same calculational scheme to Zn_0.75Cr_0.25Te shows that, in good correlation with experimental data, this system does not possess charge carriers and is characterized by strong antiferromagnetic exchange interactions. Comparing (ZnCr)Te with III-V DMS, we note the role of a large semiconducting gap for preserving high spin polarization of the states at the Fermi level at finite temperatures.