Optical excitations in hexagonal nanonetwork materials, for example, Boron-Nitride (BN) sheets and nanotubes, are investigated theoretically. The bonding of BN systems is positively polarized at the B site, and is negatively polarized at the N site. There is a permanent electric dipole moment along the BN bond, whose direction is from the B site to the N site. When the exciton hopping integral is restricted to the nearest neighbors, the flat band of the exciton appears at the lowest energy. The higher optical excitations have excitation bands similar to the electronic bands of graphene planes and carbon nanotubes. The symmetry of the flat exciton band is optically forbidden, indicating that the excitons related to this band will show quite long lifetime which will cause strong luminescence properties.