We examine the evolution of magnetic properties in the normal spinel oxides Mg(1-x)Cu(x)Cr2O4 using magnetization and heat capacity measurements. The end-member compounds of the solid solution series have been studied in some detail because of their very interesting magnetic behavior. MgCr2O4 is a highly frustrated system that undergoes a first-order structural transition at its antiferromagnetic ordering temperature. CuCr2O4 is tetragonal at room temperature as a result of Jahn-Teller active tetrahedral Cu2+ and undergoes a magnetic transition at 135 K. Substitution of magnetic cations for diamagnetic Mg2+ on the tetrahedral A site in the compositional series Mg(1-x)Cu(x)Cr2O4 dramatically affects magnetic behavior. In the composition range 0 ≤ x ≤ ≈0.3, the compounds are antiferromagnetic. A sharp peak observed at 12.5 K in the heat capacity of MgCr2O4 corresponding to a magnetically driven first-order structural transition is suppressed even for small x. Uncompensated magnetism--with open magnetization loops--develops for samples in the x range ≈0.43 ≤ x ≤ 1. Multiple magnetic ordering temperatures and large coercive fields emerge in the intermediate composition range 0.43 ≤ x ≤ 0.47. The Néel temperature increases with increasing x across the series while the value of the Curie-Weiss Θ(CW) decreases. A magnetic temperature-composition phase diagram of the solid solution series is presented.