When automatically generating a sentence description for an image or video, it often remains unclear how well the generated caption is grounded, that is whether the model uses the correct image regions to output particular words, or if the model is hallucinating based on priors in the dataset and/or the language model. The most common way of relating image regions with words in caption models is through an attention mechanism over the regions that are used as input to predict the next word. The model must therefore learn to predict the attentional weights without knowing the word it should localize. This is difficult to train without grounding supervision since recurrent models can propagate past information and there is no explicit signal to force the captioning model to properly ground the individual decoded words. In this work, we help the model to achieve this via a novel cyclical training regimen that forces the model to localize each word in the image after the sentence decoder generates it, and then reconstruct the sentence from the localized image region(s) to match the ground-truth. Our proposed framework only requires learning one extra fully-connected layer (the localizer), a layer that can be removed at test time. We show that our model significantly improves grounding accuracy without relying on grounding supervision or introducing extra computation during inference, for both image and video captioning tasks. Code is available at https://github.com/chihyaoma/cyclical-visual-captioning.