Optical transmission properties of periodic gold nanoring arrays fabricated on a free-standing silicon nitride membrane have been investigated experimentally and numerically. The transmission dips caused by the localized surface plasmon resonance (LSPR) are observed and the simulated results show pretty good agreement with the experimental data. The analysis of the electric field distributions indicates the existence of symmetric bonding modes and asymmetric anti-bonding modes of LSPR in the gold nanoring arrays. In addition, the electric field may be enhanced not only at the center of the gold nanorings but also in the spaces between the rings. Moreover, a transition from the anti-bonding mode to bonding mode is achieved by increasing the thickness of the ring. This phenomenon could be interpreted by the effect of linewidth-thickness ratio successfully. These properties may provide valuable and promising applications of advanced LSPR optical devices.