We consider the problem of the detection of brain hemorrhages from three-dimensional (3D) electrical impedance tomography (EIT) measurements. This is a condition requiring urgent treatment for which EIT might provide a portable and quick diagnosis. We employ two neural network architectures - a fully connected and a convolutional one - for the classification of hemorrhagic and ischemic strokes. The networks are trained on a dataset with $40000$ samples of synthetic electrode measurements generated with the complete electrode model on realistic heads with a 3-layer structure. We consider changes in head anatomy and layers, electrode position, measurement noise and conductivity values. We then test the networks on several datasets of unseen EIT data, with more complex stroke modeling (different shapes and volumes), higher levels of noise and different amounts of electrode misplacement. On most test datasets we achieve $\ge 90\mathrm{\%}$ average accuracy with fully connected neural networks, while the convolutional ones display an average accuracy $\ge 80\mathrm{\%}$. Despite the use of simple neural network architectures, the results obtained are very promising and motivate the applications of EIT-based classification methods on real phantoms and ultimately on human patients.