Collective properties of the low-lying levels in the odd-A (67-73)Cu were investigated by Coulomb excitation with radioactive beams. The beams were produced at ISOLDE and postaccelerated by REX-ISOLDE up to 2.99 MeV/u. In (67,69)Cu, low-lying 1/2(-), 5/2(-), and 7/2(-) states were populated. In (71,73)Cu, besides the known transitions deexciting the single-particle-like 5/2(-) and core-coupled 7/2(-) levels, gamma rays of 454 and 135 keV, respectively, were observed for the first time. Based on a reanalysis of beta-decay work and comparison with the systematics, a spin 1/2(-) is suggested for these excited states. Three B(E2) values were determined in each of the four isotopes. The results indicate a significant change in the structure of the odd-A Cu isotopes beyond N=40 where single-particle-like and collective levels are suggested to coexist at very low excitation energies.