How should we gather information to make effective decisions? A classical answer to this fundamental problem is given by the decision-theoretic value of information. Unfortunately, optimizing this objective is intractable, and myopic (greedy) approximations are known to perform poorly. In this paper, we introduce DIRECT, an efficient yet near-optimal algorithm for nonmyopically optimizing value of information. Crucially, DIRECT uses a novel surrogate objective that is: (1) aligned with the value of information problem (2) efficient to evaluate and (3) adaptive submod- ular. This latter property enables us to utilize an efficient greedy optimization while providing strong approximation guarantees. We demonstrate the utility of our approach on four diverse case-studies: touch-based robotic localization, comparison-based preference learning, wild-life conserva- tion management, and preference elicitation in behavioral economics. In the first application, we demonstrate DIRECT in closed-loop on an actual robotic platform.