Crystal structures of three new lithium copper borates, Li3CuB3O7, Li8Cu7B14O32, and Li2Cu9B12O28, and a new Li6CuB4O10 polymorph were solved by single-crystal X-ray diffraction. In all of the structures, the boron cations form BO3 triangles, which are connected with each other and with copper polyhedra only via corners in Li6CuB4O10 and Li3CuB3O7 and via both corners and edges in Li8Cu7B14O32 and Li2Cu9B12O28. The Li3CuB3O7 and Li8Cu7B14O32 compounds were synthesized as pure samples with only trace amounts of impurities; hence, their magnetic properties could be investigated and analyzed in terms of underlying magnetic couplings. Other compositions always represented multiphase mixtures. Li3CuB3O7 features infinite Cu,O chains formed by Cu2O6 units consisting of edge-shared CuO4 squares. Together with two apical oxygen atoms with long interatomic Cu-O distances of 2.7-2.8 angstrom, the Cu2O6 units form chains extended along the a-axis. These pseudochains are responsible for strong anisotropic thermal expansion behavior. The temperature dependence of the magnetization between 4 and 380 K for Li3CuB3O7 could be fit well by a spin-dimer model. The magnetic susceptibility of Li8Cu7B14O32 showed a more complex temperature dependence, with two different Curie-Weiss regimes in the temperature range of 2-380 K.