Recently, micro-video sharing platforms such as Kuaishou and Tiktok have become a major source of information for people's lives. Thanks to the large traffic volume, short video lifespan and streaming fashion of these services, it has become more and more pressing to improve the existing recommender systems to accommodate these challenges in a cost-effective way. In this paper, we propose a novel concept-aware denoising graph neural network (named CONDE) for micro-video recommendation. CONDE consists of a three-phase graph convolution process to derive user and micro-video representations: warm-up propagation, graph denoising and preference refinement. A heterogeneous tripartite graph is constructed by connecting user nodes with video nodes, and video nodes with associated concept nodes, extracted from captions and comments of the videos. To address the noisy information in the graph, we introduce a user-oriented graph denoising phase to extract a subgraph which can better reflect the user's preference. Despite the main focus of micro-video recommendation in this paper, we also show that our method can be generalized to other types of tasks. Therefore, we also conduct empirical studies on a well-known public E-commerce dataset. The experimental results suggest that the proposed CONDE achieves significantly better recommendation performance than the existing state-of-the-art solutions.