X-ray absorption spectroscopy (XAS) at the Cu K-edge is an important tool for probing the properties of copper centers in transition-metal chemistry and catalysis. However, the interpretation of experimental XAS spectra requires a detailed understanding of the dependence of spectroscopic features on the local geometric and electronic structure, which can be established by theoretical X-ray spectroscopy. Here, we present a systematic computational study of the Cu K-edge XAS spectra of selected Cu complexes based on time-dependent density-functional theory in combination with a molecular orbital analysis of the relevant transitions. For a series of Cu ammine model complexes as well as a comprehensive test set of 12 Cu(I) and 5 Cu(II) complexes, we revisit the dependence of the pre-edge region in Cu K-edge XAS spectra on oxidation state and coordination geometry. While our calculations confirm earlier experimental assignments, we can also reveal additional signatures of the ligand orbitals and identify the underlying orbital interactions. The comprehensive picture revealed by this study will provide a reliable basis for the interpretation of in situ Cu K-edge XAS spectra of catalytic intermediates.