Electro-optic tuning by using lithium niobate (LiNbO3) possesses the unique features of sub-picosecond response time, low power consumption, reduced thermal noise, and combination of nonlinear-optic applications. Here, we demonstrate a new device design and fabrication to realize on-chip electro-optically tunable LiNbO3 microresonators with high performance. This device design solves the problem of cancel-out of effective index variation induced by electro-optic effect in the microresonators made of anisotropic materials. Besides, it enables the device functions of high quality factor, polarization-selective electro-optic tuning, and large free spectral range simultaneously with few high-order modes. The microdisk structure with the air-bridge wiring produced by the standard semiconductor process facilitates the realization of integrated photonic circuits. The presented device achieves the electro-optic tuning rate up to 29.2 pm/V and the large free spectral range of 17.1 nm. The proposed device design offers highly-efficient platforms for nonlinear optics, cavity optomechanics, Kerr frequency comb generation, and cavity quantum electrodynamics.