Lithium niobate (LN), dubbed by many as the silicon of photonics, has recently risen to the forefront of chip-scale nonlinear optics research since its demonstration as an ultralow-loss integrated photonics platform. Due to its significant quadratic nonlinearity ($chi^{(2)}$), LN inspires many important applications such as second-harmonic generation (SHG), spontaneous parametric down-conversion, and optical parametric oscillation. Here, we demonstrate high-efficiency SHG in dual-resonant, periodically poled z-cut LN microrings, where quasi-phase matching is realized by field-assisted domain engineering. Meanwhile, dual-band operation is accessed by optimizing the coupling conditions in fundamental and second-harmonic bands via a single pulley waveguide. As a result, when pumping a periodically poled LN microring in the low power regime at around 1617nm, an on-chip SHG efficiency of 250,000%/W is achieved, a state-of-the-art value reported among current integrated photonics platforms. An absolute conversion efficiency of 15% is recorded with a low pump power of 115$mu$W in the waveguide. Such periodically poled LN microrings also present a versatile platform for other cavity-enhanced quasi-phase matched $chi^{(2)}$ nonlinear optical processes.