The impulse response of wireless channels between each transmit and receive antenna in a MIMO-OFDM system is known to be approximately sparse, in the sense that it has a small number of significant components relative to the channel delay spread. Moreover, it is known that the channel impulse responses in a MIMO-OFDM system are approximately group-sparse (a-group-sparse), i.e., the time-lags of the significant paths of channel impulse response between every transmit and receive antenna pair coincide. Accordingly, we cast the problem of estimating the a-group-sparse channels in the Bayesian framework, and propose novel algorithms that employ the multiple measurement vectors at the N_r receive antennas. First, we adapt the known MSBL algorithm for pilot-based a-group-sparse channel estimation in MIMO-OFDM systems. Subsequently, we generalize the MSBL algorithm to obtain a novel J-MSBL algorithm for joint a-group-sparse channel estimation and data detection. We illustrate the efficacy of the proposed techniques in terms of the mean square error and coded bit error rate performance using Monte Carlo simulations.