Modern computer systems store vast amounts of personal data, enabling advances in AI and ML but risking user privacy and trust. For privacy reasons, it is desired sometimes for an ML model to forget part of the data it was trained on. This paper presents a new approach to machine unlearning using forgetting neural networks (FNN). FNNs are neural networks with specific forgetting layers, that take inspiration from the processes involved when a human brain forgets. While FNNs had been proposed as a theoretical construct, they have not been previously used as a machine unlearning method. We describe four different types of forgetting layers and study their properties. In our experimental evaluation, we report our results on the MNIST handwritten digit recognition and fashion datasets. The effectiveness of the unlearned models was tested using Membership Inference Attacks (MIA). Successful experimental results demonstrate the great potential of our proposed method for dealing with the machine unlearning problem.