The growing market for smart home IoT devices promises new conveniences for consumers while presenting new challenges for preserving privacy within the home. Many smart home devices have always-on sensors that capture users' offline activities in their living spaces and transmit information about these activities on the Internet. In this paper, we demonstrate that an ISP or other network observer can infer privacy sensitive in-home activities by analyzing Internet traffic from smart homes containing commercially-available IoT devices even when the devices use encryption. We evaluate several strategies for mitigating the privacy risks associated with smart home device traffic, including blocking, tunneling, and rate-shaping. Our experiments show that traffic shaping can effectively and practically mitigate many privacy risks associated with smart home IoT devices. We find that 40KB/s extra bandwidth usage is enough to protect user activities from a passive network adversary. This bandwidth cost is well within the Internet speed limits and data caps for many smart homes.