Three novel metal-organic frameworks (MOFs), [Cu(1)SO4].H2O (4), [Cu2(2)2(SO4)2].4H2O (5) and [Cu(3)(H2O)]SO4.5.5H2O (6), were obtained by hydrothermal reactions of CuSO4.5H2O with the corresponding ligands, which have different flexibility. The structures of the synthesized complexes were determined by single-crystal X-ray diffraction analyses. Complex 4 has a 2D network structure with two types of metallacycles. Complex 5 also has a 2D network structure in which each independent 2D sheet contains two sub-layers bridged by oxygen atoms of the sulfate anions. Complex 6 has a 2D puckered structure in which the sulfate anions serve as counter anions, which are different from those in complexes 4 (terminators) and 5 (bridges). The different structures of complexes 4, 5 and 6 indicate that the nature of organic ligands affected the structures of the assemblies greatly. The magnetic behavior of complex 5 and anion-exchange properties of complex 6 were investigated.