Peer-to-peer live streaming systems allow a bandwidth-constrained source to broadcast a video feed to a large number of users. In addition, a design with high link utilization can achieve high stream rates, supporting high-quality video. Until now, only tree-based designs have been shown to achieve close-to-optimal rates in real-life conditions, leaving the question open as to the attainable efficiency of completely unstructured mesh-based approaches. In this paper we answer that question by showing that a carefully-designed mesh-based system can achieve close-to-optimal stream rates. Specifically, we implement and evaluate a design based on a mesh-based algorithm called DP/LU. Contrary to tree-based designs, DP/LU uses an unstructured overlay, which is easier to construct and is highly resistant to churn. In addition, we introduce mechanisms for overlay rewiring and source scheduling that lead to significant performance improvements. Our experimental evaluation shows that our design achieves 95% of the maximum achievable stream rate in a static environment, and 90% under high churn. This demonstrates that mesh-based designs are an excellent choice for scalable and robust high-quality peer-to-peer live streaming.