The modeling of conversational context plays a vital role in emotion recognition from conversation (ERC). In this paper, we put forward a novel idea of encoding the utterances with a directed acyclic graph (DAG) to better model the intrinsic structure within a conversation, and design a directed acyclic neural network, namely DAG-ERC, to implement this idea. In an attempt to combine the strengths of conventional graph-based neural models and recurrence-based neural models, DAG-ERC provides a more intuitive way to model the information flow between long-distance conversation background and nearby context. Extensive experiments are conducted on four ERC benchmarks with state-of-the-art models employed as baselines for comparison. The empirical results demonstrate the superiority of this new model and confirm the motivation of the directed acyclic graph architecture for ERC.