As a widely applied model for compressive sensing, the multitask compressive sensing can improve the performance of the inversion by appropriately exploiting the interrelationships of the tasks. The existing multitask compressive sensing recovery algorithms only utilize the statistical characteristics of a sparse signal, the structural characteristics of the sparse signal have not been taken into consideration. A multitask compressive sensing recovery algorithm is proposed in this paper based on the block sparse Bayesian learning. The block sparse single measurement vector model is applied to the multi-task problem. Both statistical and block structural characteristics of the sparse signal are used to build a mathematical model, and the sparse inverse problem is linked to the parameter iteration problems in the Bayesian framework. The proposed algorithm does not require the sparseness information and noise beforehand, which turns out to be an effective blind recovery algorithm. Extensive numerical experiments show that the proposed algorithm can exploit both statistical and structural characteristics of the signal, therefore it may reach a good trade-off between the recovery accuracy and the convergence rate.