Target search with unmanned aerial vehicles (UAVs) is relevant problem to many scenarios, e.g., search and rescue (SaR). However, a key challenge is planning paths for maximal search efficiency given flight time constraints. To address this, we propose the Obstacle-aware Adaptive Informative Path Planning (OA-IPP) algorithm for target search in cluttered environments using UAVs. Our approach leverages a layered planning strategy using a Gaussian Process (GP)based model of target occupancy to generate informative paths in continuous 3D space. Within this framework, we introduce an adaptive replanning scheme which allows us to trade off between information gain, field coverage, sensor performance, and collision avoidance for efficient target detection. Extensive simulations show that our OA-IPP method performs better than state-of-the-art planners, and we demonstrate its application in a realistic urban SaR scenario.