This paper presents a novel iterative Bayesian algorithm, Block Iterative Bayesian Algorithm (Block-IBA), for reconstructing block-sparse signals with unknown block structures. Unlike the other existing algorithms for block sparse signal recovery which assume the cluster structure of the non-zero elements of the unknown signal to be independent and identically distributed (i.i.d.), we use a more realistic Bernoulli-Gaussian hidden Markov model (BGHMM) to capture the burstiness (block structure) of the impulsive noise in practical applications such as Power Line Communication (PLC). The Block-IBA iteratively estimates the amplitudes and positions of the block-sparse signal based on Expectation-Maximization (EM) algorithm which is also optimized with the steepest-ascent method. Simulation results show the effectiveness of our algorithm for block-sparse signal recovery.Index Terms-Block-sparse, iterative Bayesian algorithm, steepest-ascent.