Cells can communicate with each other by emitting diffusible signaling molecules into the surrounding environment. However, simple diffusion is slow. Even small molecules take hours to diffuse millimeters away from their source, making it difficult for thousands of cells to coordinate their activity over millimeters, as happens routinely during development and immune response. Moreover, simple diffusion creates shallow, Gaussian-tailed concentration profiles. Attempting to move up or down such shallow gradients - to chemotax - is a difficult task for cells, as they see only small spatial and temporal concentration changes. Here, we demonstrate that cells utilizing diffusive relays, in which the presence of one type of diffusible signaling molecule triggers participation in the emission of the same type of molecule, can propagate fast-traveling diffusive waves that give rise to steep chemical gradients. Our methods are general and capture the effects of dimensionality, cell density, signaling molecule degradation, pulsed emission, and cellular chemotaxis on the diffusive wave dynamics. We show that system dimensionality - the size and shape of the extracellular medium and the distribution of the cells within it - can have a particularly dramatic effect on wave initiation and asymptotic propagation, and that these dynamics are insensitive to the details of cellular activation. As an example, we show that neutrophil swarming experiments exhibit dynamical signatures consistent with the proposed signaling motif. Interpreted in the context of these experiments, our results provide insight into the utility of signaling relays in immune response.