Aspect sentiment triplet extraction (ASTE), which aims to identify aspects from review sentences along with their corresponding opinion expressions and sentiments, is an emerging task in fine-grained opinion mining. Since ASTE consists of multiple subtasks, including opinion entity extraction, relation detection, and sentiment classification, it is critical and challenging to appropriately capture and utilize the associations among them. In this paper, we transform ASTE task into a multi-turn machine reading comprehension (MTMRC) task and propose a bidirectional MRC (BMRC) framework to address this challenge. Specifically, we devise three types of queries, including non-restrictive extraction queries, restrictive extraction queries and sentiment classification queries, to build the associations among different subtasks. Furthermore, considering that an aspect sentiment triplet can derive from either an aspect or an opinion expression, we design a bidirectional MRC structure. One direction sequentially recognizes aspects, opinion expressions, and sentiments to obtain triplets, while the other direction identifies opinion expressions first, then aspects, and at last sentiments. By making the two directions complement each other, our framework can identify triplets more comprehensively. To verify the effectiveness of our approach, we conduct extensive experiments on four benchmark datasets. The experimental results demonstrate that BMRC achieves state-of-the-art performances.