A concrete block system is provided and allows for the use of a single type of non-integral connector to perform different functions in a mortarless structure, the system comprising a plurality of structural concrete blocks and a plurality of veneer units. The same connector can be used to connect veneer units to structural blocks, to retain the veneer units in different orientations; to create near vertical installations or a setback between stacked rows of structural blocks. In addition, the system provides a single structural block, which can be used for both straight portions or corners of a same structure. The structural block can also be used to create double-sided walls, with opposite lateral walls being configured to received/retain veneer units thereon.