It is a fundamental issue to achieve efficient information collection in Internet of Things (IoT), where random (channel) access plays an indispensable role, especially when coordination among IoT end nodes is unachievable. Compressive sensing (CS) has been widely used in random access to facilitate energy efficient and accurate data collection. However, a joint sparsity structure, which commonly exists among signals acquired by different end nodes, has been long ignored by existing CS-based random access schemes, leading to insufficient energy efficiency and accuracy. In this article, capitalizing on this joint sparsity structure, we propose a structured compressive random access (SCRA) mechanism in order to achieve maximum energy efficiency with accuracy guarantee for data collection. Specifically, we first model the data loss induced by packet collisions during random access as an independent CS measurement process for each node, where the corresponding CS projection matrix is determined by the data loss pattern. Furthermore, in order to control the amount of data transmitted in the channel and alleviate the packet collisions, we employ the concept of sensing probability to perform random subsampling at each end node before transmission, where the optimal sensing probability is derived. Finally, we propose to jointly recover the set of original signals at all nodes based on the concept of group sparsity by formulating the data collection process as a single-measurement-vector problem in CS. The evaluation results validate the effectiveness of SCRA in utilizing the joint sparsity structure to obtain superior performance compared to the benchmark methods.