Aspect Sentiment Triplet Extraction (ASTE) is a complex and challenging task in Natural Language Processing (NLP). It aims to extract the triplet of aspect term, opinion term, and their associated sentiment polarity, which is a more fine-grained study in Aspect Based Sentiment Analysis. Furthermore, there have been a large number of approaches being proposed to handle this relevant task. However, existing methods for ASTE suffer from powerless interactions between different sources of textual features, and they usually exert an equal impact on each type of feature, which is quite unreasonable while building contextual representation. Therefore, in this paper, we propose a novel Multi-Branch GCN (MBGCN)-based ASTE model to solve this problem. Specifically, our model first generates the enhanced semantic features via the structure-biased BERT, which takes the position of tokens into the transformation of self-attention. Then, a biaffine attention module is utilized to further obtain the specific semantic feature maps. In addition, to enhance the dependency among words in the sentence, four types of linguistic relations are defined, namely part-of-speech combination, syntactic dependency type, tree-based distance, and relative position distance of each word pair, which are further embedded as adjacent matrices. Then, the widely used Graph Convolutional Network (GCN) module is utilized to complete the work of integrating the semantic feature and linguistic feature, which is operated on four types of dependency relations repeatedly. Additionally, an effective refining strategy is employed to detect whether word pairs match or not, which is conducted after the operation of each branch GCN. At last, a shallow interaction layer is designed to achieve the final textual representation by fusing the four branch features with different weights. To validate the effectiveness of MBGCNs, extensive experiments have been conducted on four public and available datasets. Furthermore, the results demonstrate the effectiveness and robustness of MBGCNs, which obviously outperform state-of-the-art approaches.