An optimized implementation of a block tridiagonal solver based on the block cyclic reduction (BCR) algorithm is introduced and its portability to graphics processing units (GPUs) is explored. The computations are performed on the NVIDIA GTX480 GPU. The results are compared with those obtained on a single core of Intel Core i7-920 (2.67 GHz) in terms of calculation runtime. The BCR linear solver achieves the maximum speedup of 5.84x with block size of 32 over the CPU Thomas algorithm in double precision. The proposed BCR solver is applied to discontinuous Galerkin (DG) simulations on structured grids via alternating direction implicit (ADI) scheme. The GPU performance of the entire computational fluid dynamics (CFD) code is studied for different compressible inviscid flow test cases. For a general mesh with quadrilateral elements, the ADI-DG solver achieves the maximum total speedup of 7.45x for the piecewise quadratic solution over the CPU platform in double precision.