We propose a novel design of photonic-crystal nanobeam cavities in lithium niobate (LN) for both TE and TM modes, addressing problems associated with tilted sidewalls, which commonly result from dry etching. Using optimized periodical nano-groove structures, the proposed devices could achieve quality factors as high as 3.9×10(6) with a modal volume of 5.0 (λ/n)3. We also show that such a design is promising for applications in nonlinear optics by theoretically predicting the efficiencies for electro-optic modulation and second-harmonic generation. The proposed nano-groove structures and design rules could also be applied for other material platforms that possess tilted sidewalls.