In this Letter, we investigate the formation of trapped modes with near-zero group velocities in a ring chain composed of dielectric nanorods. Two kinds of bound modes are successfully identified: the regular below-continuum-resonance (BCR) modes formed at the band edge and importantly the quasi-bound-states-in-the-continuum (BIC) trapped modes (similar to the BIC in the equivalent infinite linear chain). The lowest-order trapped mode possesses the highest Q factor, which scales exponentially with the number of nanorods N as Q∼exp(0.325N) for the BCR and Q∼exp(0.662N) for the quasi-BIC. Interestingly, a moderate high Q factor ∼105 can be obtained for the quasi-BIC mode even with a very small N=8. This suggests that our nanorod-based ring resonator possesses a clear advantage over the linear chain for the same N. Our findings greatly expand the application scope of BIC-based phenomena.