Searching for food, friends, and mates often begins with an airborne scent. Importantly, odor concentration rises with physical proximity to an odorous source, suggesting a framework for orienting within olfactory landscapes to optimize behavior. Here, we created a two-dimensional odor space composed purely of odor stimuli to model how a navigator encounters smells in a natural environment. We show that human subjects can learn to navigate in olfactory space and form predictions of to-be-encountered smells. During navigation, fMRI responses in entorhinal cortex and ventromedial prefrontal cortex take the form of grid-like representations with hexagonal periodicity and entorhinal grid strength scaled with behavioral performance across subjects. The identification of olfactory grid-like codes with 6-fold symmetry highlights a unique neural mechanism by which odor information can be assembled into spatially navigable cognitive maps, optimizing orientation, and path finding toward an odor source.