We present an efficient three-dimensional (3D) time-domain airborne electromagnetic (EM) modeling based on finite-volume method, in combination with the merits of solving the secondary field, local mesh, and the direct solver. Taking the strategy of separating primary from secondary field for the calculation of time-domain EM field, we greatly reduce the number of grids in comparison to direct solution of the total field. The primary field is obtained by Hankel transform from the frequency-domain for a half-space or layered earth model. The techniques of local mesh and direct solver are adopted to further improve the modeling efficiency. We compare and discuss the characteristics of the method developed in this paper from time consumption and grid sizes. The EM responses for different transmitting waveforms are calculated via a convolution between step-wave responses and the transmitting current or its derivatives. All numerical experiments on models with multiple geological bodies and fracture zones have demonstrated that our modeling algorithm is efficient and effective when simulating AEM system responses.