We propose and demonstrate a two-ring coupled Mach-Zehnder interferometer (2RMZI) device that exhibits a sharp resonance with background suppression to give both high finesse and modulation depth. The combination of MZI and the two-ring structure leads to more complete destructive interference that effectively removes the unwanted background envelope effect found in the transmission spectrum of a simple two-ring-two-bus (2R2B) system. The projected finesse enhancement of 2RMZI relative to one-ring coupled MZI and 2R2B is discussed based on best-fit parameter values that match the fabricated devices.