We studied resistance oscillations in two types of superconducting mesoscopic Nb rings. In a simple superconducting ring device, a resistance oscillation with a period of the quantized magnetic flux h/2e was clearly observed. On the other hand, in a micro-bridge ring device where two-narrow parts are embedded in parallel and work as superconductor-normal metal-superconductor junctions, higher harmonic resistance oscillations were obtained when the measurement current was well-tuned. We argue that such higher harmonic resistance oscillations can be detected even in the micro-bridge Nb superconducting ring device where the device size is much larger than the coherence length of Nb.