This paper discusses techniques for accelerated computation of several fast spectral transforms on graphics processing units (GPUs) using the Open Computing Language (OpenCL). We present a reformulation of fast algorithms which takes into account peculiar properties of transforms to make them suitable for the GPU implementation. A special attention is paid to the organization of computations, memory transfer reductions, impact of integer and Boolean arithmetic, different structure of algorithms, etc. Performance of the GPU implementations is compared with the classical C/C++ implementations for the central processing unit (CPU). Experiments confirm that, even though the spectral transforms considered involve only simple arithmetic, significant speedups are achieved by implementing the algorithms in OpenCL and performing them on the GPU.