New coordination compounds of copper(II) with 2,5-bis(ethylthio)-1,3,4-thiadiazole (L1) and 2,5-bis(pyridylmethylthio)-1,3,4-thiadiazole (L2) with compositions Cu(L1)2Br2, Cu(L1)(C2N3)2, Cu(L2)Cl2, and Cu(L2)Br2 were prepared. The complexes were identified and studied by CHN analysis, infrared (IR) spectroscopy, powder X-Ray diffraction (XRD), and static magnetic susceptibility. The crystal structures of Cu(II) complexes with L1 were determined. The structures of the coordination core of complexes Cu(L2)Cl2 and Cu(L2)Br2 were determined by Extended X-ray absorption fine structure (EXAFS) spectroscopy. Magnetization measurements have revealed various magnetic states in the studied complexes, ranging from an almost ideal paramagnet in Cu(L1)2Br2 to alternating-exchange antiferromagnetic chains in Cu(L1)(C2N3)2, where double dicyanamide bridges provide an unusually strong exchange interaction (J1/kB ≈ −23.5 K; J2/kB ≈ −20.2 K) between Cu(II) ions. The cytotoxic activity of copper(II) complexes with L2 was estimated on the human cell lines of breast adenocarcinoma (MCF-7) and hepatocellular carcinoma (HepG2).