A straightforward implementation of multi-block lattice Boltzmann method (MB-LBM) on a graphical processing unit (GPU) is presented to accelerate simulations of complex fluid flows. The characteristics of MB-LBM algorithm are analyzed in detail. The algorithm is tested in terms of accuracy and computational time with the benchmark cases of lid driven cavity flow and the flow past a circular cylinder, and satisfactory results are obtained. The results show the performance on GPU is consistently better than that on CPU, and the greater the amount of data, the larger the acceleration ratio. Moreover, the arrangement of computational domain has significant effects on the performance of GPU. These results demonstrate the great potential of GPU on MB-LBM, especially for the calculation with large amounts of data.