Animal-Human-Machine (AHM) teams are a type of hybrid intelligence system wherein interactions between a human, AI-enabled machine, and animal members can result in unique capabilities greater than the sum of their parts. This paper calls for a systematic approach to studying the design of AHM team structures to optimize performance and overcome limitations in various applied settings. We consider the challenges and opportunities in investigating the synergistic potential of AHM team members by introducing a set of dimensions of AHM team functioning to effectively utilize each member's strengths while compensating for individual weaknesses. Using three representative examples of such teams -- security screening, search-and-rescue, and guide dogs -- the paper illustrates how AHM teams can tackle complex tasks. We conclude with open research directions that this multidimensional approach presents for studying hybrid human-AI systems beyond AHM teams.