At the radioactive ion beam facility REX-ISOLDE, neutron-rich zinc isotopes were investigated using low-energy Coulomb excitation. These experiments have resulted in B(E2, 2(1)(+)-> 0(1)(+)) values in (74-80)Zn, B(E2, 4(1)(+)-> 2(1)(+)) values in (74,76)Zn and the determination of the energy of the first excited 2(1)(+) states in (78,80)Zn. The zinc isotopes were produced by high-energy proton- (A = 74, 76, 80) and neutron-(A = 78) induced fission of (238)U, combined with selective laser ionization and mass separation. The isobaric beam was postaccelerated by the REX linear accelerator and Coulomb excitation was induced on a thin secondary target, which was surrounded by the MINIBALL germanium detector array. In this work, it is shown how the selective laser ionization can be used to deal with the considerable isobaric beam contamination and how a reliable normalization of the experiment can be achieved. The results for zinc isotopes and the N = 50 isotones are compared to collective model predictions and state-of-the-art large-scale shell-model calculations, including a recent empirical residual interaction constructed to describe the present experimental data up to 2004 in this region of the nuclear chart.