Motion planning under uncertainty is an important problem in robotics. Although probabilistic sampling is highly successful for motion planning of robots with many degrees of freedom, sampling-based algorithms typically ignore uncertainty during planning. We introduce the notion of a bounded uncertainty roadmap (BURM) and use it to extend samplingbased algorithms for planning under uncertainty in environment maps. The key idea of our approach is to evaluate uncertainty, represented by collision probability bounds, at multiple resolutions in different regions of the configuration space, depending on their relevance for finding a best path. Preliminary experimental results show that our approach is highly effective: our BURM algorithm is at least 40 times faster than an algorithm that tries to evaluate collision probabilities exactly, and it is not much slower than classic probabilistic roadmap planning algorithms, which ignore uncertainty in environment maps.