Thermal simulations are an important part in the design of electronic systems, especially as systems with high power density become common. In simulation-based design approaches, a considerable amount of time is spent by repeated simulations. In this work, we present a proof-of-concept study of the application of convolutional neural networks to accelerate those thermal simulations. The goal is not to replace standard simulation tools but to provide a method to quickly select promising samples for more detailed investigations. Based on a training set of randomly generated circuits with corresponding Finite Element solutions, the full 3D steady-state temperature field is estimated using a fully convolutional neural network. A custom network architecture is proposed which captures the long-range correlations present in heat conduction problems. We test the network on a separate dataset and find that the mean relative error is around 2 % and the typical evaluation time is 35 ms per sample ( 2 ms for evaluation, 33 ms for data transfer). The benefit of this neural-network-based approach is that, once training is completed, the network can be applied to any system within the design space spanned by the randomised training dataset (which includes different components, material properties, different positioning of components on a PCB, etc.).