Terahertz filter can be used for various applications such as imaging, spectroscopy and sensing. We propose and fabricate a terahertz multiband filter using three nested closed-ring resonators (CRRs) on polyimide substrate. We numerically analyze the physical origin of the multispectral response and discuss the dependence of the resonance frequencies on the geometrical parameters of the closed rings. It is found that the three resonant peaks of the proposed filter originate from not only the dipole electric response of the three rings and the magnetic response between the adjacent rings, but also the effect from inner or outer CRR. The performance of the filter is measured by terahertz time-domain spectroscopy (TDS), and the experimental results show three distinct resonant peaks at 0.33, 0.68, 1.13 THz, which are in good agreement with simulations.