Social robots, conversational agents, voice assistants, and other embodied AI are increasingly a feature of everyday life. What connects these various types of intelligent agents is their ability to interact with people through voice. Voice is becoming an essential modality of embodiment, communication, and interaction between computer-based agents and end-users. This survey presents a meta-synthesis on agent voice in the design and experience of agents from a human-centered perspective: voice-based human--agent interaction (vHAI). Findings emphasize the social role of voice in HAI as well as circumscribe a relationship between agent voice and body, corresponding to human models of social psychology and cognition. Additionally, changes in perceptions of and reactions to agent voice over time reveals a generational shift coinciding with the commercial proliferation of mobile voice assistants. The main contributions of this work are a vHAI classification framework for voice across various agent forms, contexts, and user groups, a critical analysis grounded in key theories, and an identification of future directions for the oncoming wave of vocal machines.