A critical step in understanding how a genome functions is determining which transcription factors (TFs) regulate each gene. Accordingly, extensive effort has been devoted to mapping TF networks. In Saccharomyces cerevisiae, protein-DNA interactions have been identified for most TFs by ChIP-chip, and expression profiling has been done on strains deleted for most TFs. These studies revealed that there is little overlap between the genes whose promoters are bound by a TF and those whose expression changes when the TF is deleted, leaving us without a definitive TF network for any eukaryote and without an efficient method for mapping functional TF networks. This paper describes NetProphet, a novel algorithm that improves the efficiency of network mapping from gene expression data. NetProphet exploits a fundamental observation about the nature of TF networks: The response to disrupting or overexpressing a TF is strongest on its direct targets and dissipates rapidly as it propagates through the network. Using S. cerevisiae data, we show that NetProphet can predict thousands of direct, functional regulatory interactions, using only gene expression data. The targets that NetProphet predicts for a TF are at least as likely to have sites matching the TF's binding specificity as the targets implicated by ChIP. Unlike most ChIP targets, the NetProphet targets also show evidence of functional regulation. This suggests a surprising conclusion: The best way to begin mapping direct, functional TF-promoter interactions may not be by measuring binding. We also show that NetProphet yields new insights into the functions of several yeast TFs, including a well-studied TF, Cbf1, and a completely unstudied TF, Eds1.