Digital predistortion has been proved to be an effective remedy for the nonlinear distortion in radio-frequency power amplifier. Unfortunately, conventional digital predistorters (DPDs) developed for single-band transmitters can not be directly applied to concurrent dual-band. To solve the problem, DPDs with two dimensional memory polynomials were then proposed. Although they can provide good performance, its computational complexity is high for the estimation of polynomial coefficients. Look-up-table (LUT) based DPDs, developed for single-band transmitters, can effectively solve the problem. However, there are limited studies about LUT based DPDs for concurrent dual-band transmitters. In this paper, we propose a two-dimensional LUT based Hammerstein DPD to solve the problem. Analyzing the signals on both bands, we propose a new adaptive algorithm with the gradient-descent and recursive-least-squares (RLS) methods to train the DPD iteratively. With the proposed method, the computational complexity in the identification of the DPDs can be significantly reduced. Finally, simulations and experiments are conducted to demonstrate that the performance of the proposed DPD is comparable to that of the polynomial based DPD.