Unsupervised image captioning with no annotations is an emerging challenge in computer vision, where the existing arts usually adopt GAN (Generative Adversarial Networks) models. In this paper, we propose a novel memory-based network rather than GAN, named Recurrent Relational Memory Network (R2M). Unlike complicated and sensitive adversarial learning that non-ideally performs for long sentence generation, R2M implements a concepts-to-sentence memory translator through two-stage memory mechanisms: fusion and recurrent memories, correlating the relational reasoning between common visual concepts and the generated words for long periods. R2M encodes visual context through unsupervised training on images, while enabling the memory to learn from irrelevant textual corpus via supervised fashion. Our solution enjoys less learnable parameters and higher computational efficiency than GAN-based methods, which heavily bear parameter sensitivity. We experimentally validate the superiority of R2M than state-of-the-arts on all benchmark datasets.