Building upon the latest implementation of the Moving Morphable Component (MMC) method, this work presents a novel and efficient framework for solving natural frequency topology optimization problems. Unlike traditional density-based methods, the MMC method describes structural topology using explicit geometric components. This explicit description inherently facilitates the formation of smooth and crisp structural boundaries, which is particularly advantageous for natural frequency optimization problems. To effectively tackle persistent challenges in dynamic optimization, such as spurious eigenmodes in low-density regions, the non-differentiability of repeated natural frequencies, and mode switching phenomena, we develop a comprehensive set of strategies. Specifically, analytical sensitivity analyses based on the adjoint method are derived to efficiently compute gradients for both non-repeated and repeated natural frequencies. Additionally, a Modal Assurance Criterion (MAC) constraint is integrated to ensure mode tracking throughout the optimization process. Furthermore, to fully leverage the advantages of the MMC method, we propose adaptive key parameter scaling strategies and a new load transfer path identification algorithm. By utilizing connectivity analysis, this algorithm automatically identifies and retains only the degrees of freedom (DOFs) effectively connecting the structural mass to the supports. This technique serves a dual purpose: it effectively eliminates spurious eigenmodes caused by isolated weak elements and significantly accelerates the optimization process by removing redundant DOFs. Numerical examples, ranging from 2D beams to complex 3D structures, demonstrate the efficiency and robustness of the presented method. Notably, in 3D cases, the proposed DOF elimination technique reduces the average iteration time significantly, making high-resolution 3D natural frequency optimization computationally feasible. Finally, compact and extensible MATLAB codes for solving the proposed 2D and 3D topology optimization problems are provided to facilitate educational use and further research.