From artificial intelligence to network security to hardware design, it is well-known that computing research drives many important technological and societal advancements. However, less is known about the long-term career paths of the people behind these innovations. What do their careers reveal about the evolution of computing research? Which institutions were and are the most important in this field, and for what reasons? Can insights into computing career trajectories help predict employer retention? In this paper we analyze several decades of post-PhD computing careers using a large new dataset rich with professional information, and propose a versatile career network model, R^3, that captures temporal career dynamics. With R^3 we track important organizations in computing research history, analyze career movement between industry, academia, and government, and build a powerful predictive model for individual career transitions. Our study, the first of its kind, is a starting point for understanding computing research careers, and may inform employer recruitment and retention mechanisms at a time when the demand for specialized computational expertise far exceeds supply.