In orthogonal frequency-division multiplexing (OFDM) systems operating over rapidly time-varying channels, the orthogonality between subcarriers is destroyed leading to inter-carrier interference (ICI) and resulting in an irreducible error floor. In this paper, a new and low-complexity maximum a posteriori probability (MAP) detection algorithm is proposed for OFDM systems operating over rapidly time-varying multipath channels. The detection algorithm exploits the banded structure of the frequency-domain channel matrix whose bandwidth is a parameter to be adjusted according to the speed of the mobile terminal. Based on this assumption, the received signal vector is decomposed into reduced dimensional sub-observations in such a way that all components of the observation vector contributing to the symbol to be detected are included in the decomposed observation model. The data symbols are then detected by the MAP algorithm by means of a Markov chain Monte Carlo (MCMC) technique in an optimal and computationally efficient way. Computational complexity investigation as well as simulation results indicate that this algorithm has significant performance and complexity advantages over existing suboptimal detection and equalization algorithms proposed earlier in the literature.