There is a growing trend in deploying DNNs on tiny microcontroller (MCUs) to provide inference capabilities in the IoT. While prior research has explored many lightweight techniques to compress DNN models, achieving overall efficiency in model inference requires not only model optimization but also careful system resource utilization for execution. Existing studies primarily leverage arithmetic logic units (ALUs) for integer-only computations on a single CPU core. Floating-point units (FPU) and multi-core capabilities available in many existing MCUs remain underutilized. To fill this gap, we propose AdaptQNet, a novel MCU neural network system that can determine the optimal precision assignment for different layers of a DNN model. AdaptQNet models the latency of various operators in DNN models across different precisions on heterogeneous processing units. This facilitates the discovery of models that utilize FPU and multi-core capabilities to enhance capacity while adhering to stringent memory constraints. Our implementation and experiments demonstrate that AdaptQNet enables the deployment of models with better accuracy-efficiency trade-off on MCUs.