The digital predistortion (DPD) technique has been utilized in radio-frequency power amplifier linearization for its low cost and good performance. However, the conventional single-band DPD design is not applicable to the concurrent dual-band transmitters since the cross modulations are induced. The two-dimensional (2D) polynomial based DPD has been proposed as a remedy. Despite its good performance, the main drawbacks are the high computational complexity for the identification of the DPD coefficients and the high implementation complexity. It is known that the look-up-table (LUT) based method, which has been applied in conventional single-band transmitters, can effectively overcome the problems. In this paper, we consider a 2D LUT based DPD for the concurrent dual-band transmitters. First, we formulate a cost function from which the elements of the tables can be obtained through the optimization. Then, we propose an adaptive algorithm using the gradient-descent (GD) method to conduct the search. Moreover, we propose a practical method to implement the GD method which can significantly reduce the computational and implementation complexities. Finally, some measurement results are presented to demonstrate that the proposed algorithm has the satisfactory performance.