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Emotion Cause Extraction in Conversations (ECEC) aims to extract the utterances which contain the emotional cause in conversations. Most prior research focuses on modelling conversational contexts with sequential encoding, ignoring the informative interactions between utterances and conversational-specific features for ECEC. In this paper, we investigate the importance of discourse structures in handling utterance interactions and conversationspecific features for ECEC. To this end, we propose a discourse-aware model (DAM) for this task. Concretely, we jointly model ECEC with discourse parsing using a multi-task learning (MTL) framework and explicitly encode discourse structures via gated graph neural network (gated GNN), integrating rich utterance interaction information to our model. In addition, we use gated GNN to further enhance our ECEC model with conversation-specific features. Results on the benchmark corpus show that DAM outperform the state-of-theart (SOTA) systems in the literature. This suggests that the discourse structure may contain a potential link between emotional utterances and their corresponding cause expressions. It also verifies the effectiveness of conversationalspecific features. The codes of this paper will be available on GitHub.