In a wireless network, the use of cooperation among nodes can significantly improve capacity and robustness to fading. Node cooperation can take many forms, including relaying and coordinated beamforming. However, many cooperation techniques have been developed for operation in narrowband systems for high signal-to-noise ratio (SNR) applications. It is important to study how relay networks perform in a low-SNR regime, where the available degrees of freedom is large and the resulting SNR per degree of freedom is small. In this paper, we investigate the achievable rates and scaling laws of bursty amplify-and-forward relay networks in the low-SNR regime. Specifically, our results allow us to understand the effect of different system parameters on the achievable rates and scaling laws in the low-SNR regime, and highlight the role of bursty transmissions in this regime. These results entirely depend on the geographic locations of the nodes and are applicable for both fixed and random networks. We identify four scaling regimes that depend on the growth of the number of relay nodes and the increase of burstiness relative to the SNR.