We study an adaptive source seeking problem, in which a mobile robot must identify the strongest emitter(s) of a signal in an environment with background emissions. Background signals may be highly heterogeneous and can mislead algorithms that are based on receding horizon control. We propose AdaSearch, a general algorithm for adaptive source seeking in the face of heterogeneous background noise. AdaSearch combines global trajectory planning with principled confidence intervals in order to concentrate measurements in promising regions while guaranteeing sufficient coverage of the entire area. Theoretical analysis shows that AdaSearch confers gains over a uniform sampling strategy when the distribution of background signals is highly variable. Simulation experiments demonstrate that when applied to the problem of radioactive source seeking, AdaSearch outperforms both uniform sampling and a receding time horizon information-maximization approach based on the current literature. We also demonstrate AdaSearch in hardware, providing further evidence of its potential for real-time implementation.