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3D dense captioning is a recently-proposed novel task, where point clouds contain more geometric information than the 2D counterpart. However, it is also more challenging due to the higher complexity and wider variety of inter-object relations contained in point clouds. Existing methods only treat such relations as by-products of object feature learning in graphs without specifically encoding them, which leads to sub-optimal results. In this paper, aiming at improving 3D dense captioning via capturing and utilizing the complex relations in the 3D scene, we propose MORE, a Multi-Order RElation mining model, to support generating more descriptive and comprehensive captions. Technically, our MORE encodes object relations in a progressive manner since complex relations can be deduced from a limited number of basic ones. We first devise a novel Spatial Layout Graph Convolution (SLGC), which semantically encodes several first-order relations as edges of a graph constructed over 3D object proposals. Next, from the resulting graph, we further extract multiple triplets which encapsulate basic first-order relations as the basic unit, and construct several Object-centric Triplet Attention Graphs (OTAG) to infer multi-order relations for every target object. The updated node features from OTAG are aggregated and fed into the caption decoder to provide abundant relational cues, so that captions including diverse relations with context objects can be generated. Extensive experiments on the Scan2Cap dataset prove the effectiveness of our proposed MORE and its components, and we also outperform the current state-of-the-art method. Our code is available at https://github.com/SxJyJay/MORE.