Five new tetradentate ligands [NNNN] with benzimidazolyl-imine or amine nitrogen donors have been synthesized in good yields under mild conditions from easily available substrates. trans-N,N'-bis(1-Ethyl-2-benzimidazolylmethylene)cyclohexane-1,2-diimine is the best accelerating ligand in this series that supports the Cu(I)-catalyzed Ullmann N-arylation and the direct three-component azide-alkyne cycloaddition reaction to give the corresponding substituted imidazole, pyrazole, and triazole in high yields. Single-crystal X-ray diffraction analysis of its complex with CuI reveals a novel one-dimensional coordination polymer of the metal chain bridged alternately by the [NNNN] ligand and diiodides.