Multimedia content is of predominance in the modern Web era. Investigating how users interact with multimodal items is a continuing concern within the rapid development of recommender systems. The majority of previous work focuses on modeling user-item interactions with multimodal features included as side information. However, this scheme is not well-designed for multimedia recommendation. Specifically, only collaborative item-item relationships are implicitly modeled through high-order item-user-item relations. Considering that items are associated with rich contents in multiple modalities, we argue that the latent semantic item-item structures underlying these multimodal contents could be beneficial for learning better item representations and further boosting recommendation. To this end, we propose a LATent sTructure mining method for multImodal reCommEndation, which we term LATTICE for brevity. To be specific, in the proposed LATTICE model, we devise a novel modality-aware structure learning layer, which learns item-item structures for each modality and aggregates multiple modalities to obtain latent item graphs. Based on the learned latent graphs, we perform graph convolutions to explicitly inject high-order item affinities into item representations. These enriched item representations can then be plugged into existing collaborative filtering methods to make more accurate recommendations. Extensive experiments on three real-world datasets demonstrate the superiority of our method over state-of-the-art multimedia recommendation methods and validate the efficacy of mining latent item-item relationships from multimodal features.