The growing number of low-power smart devices in the Internet of Things is coupled with the concept of "Edge Computing", that is moving some of the intelligence, especially machine learning, towards the edge of the network. Enabling machine learning algorithms to run on resource-constrained hardware, typically on low-power smart devices, is challenging in terms of hardware (optimized and energy-efficient integrated circuits), algorithmic and firmware implementations. This paper presents FANN-on-MCU, an open-source toolkit built upon the Fast Artificial Neural Network (FANN) library to run lightweight and energy-efficient neural networks on microcontrollers based on both the ARM Cortex-M series and the novel RISC-V-based Parallel Ultra-Low-Power (PULP) platform. The toolkit takes multi-layer perceptrons trained with FANN and generates code targeted at execution on low-power microcontrollers either with a floating-point unit (i.e., ARM Cortex-M4F and M7F) or without (i.e., ARM Cortex M0-M3 or PULP-based processors). This paper also provides an architectural performance evaluation of neural networks on the most popular ARM Cortex-M family and the parallel RISC-V processor called Mr. Wolf. The evaluation includes experimental results for three different applications using a self-sustainable wearable multi-sensor bracelet. Experimental results show a measured latency in the order of only a few microseconds and a power consumption of few milliwatts while keeping the memory requirements below the limitations of the targeted microcontrollers. In particular, the parallel implementation on the octa-core RISC-V platform reaches a speedup of 22x and a 69% reduction in energy consumption with respect to a single-core implementation on Cortex-M4 for continuous real-time classification.