Following the reports of breakthrough performances, machine learning based applications have become very popular in the medical field. However, with the recent increase in concerns related to data privacy, and the publication of specific regulations (e.g. GDPR), the development and, thus, exploitation of deep learning based applications in clinical decision making processes, has been rendered impossible in many cases. Herein, we describe and evaluate an approach that employs Fully Homo-morphic Encryption for allowing computations to be performed on sensitive data. Specifically, the solution exploits the MORE scheme and does not disclose patient data. The chosen encryption scheme increases the runtime only marginally and, importantly, allows for operations to be performed directly on floating point numbers, which represents a critical property for artificial neural networks. The feasibility and performance are first evaluated on a standard benchmarking application (MNIST digit classification). Next, we considered a medical imaging application, i.e. classification of coronary views in X-ray angiography. The reported results indicate that the proposed solution has great potential: (i) computational results are indistinguishable from those obtained with the unencrypted variants of the deep learning based applications, and (ii) run times increase only marginally. Finally, we also discuss in detail security concerns, and emphasize that the proposed solution may be employed in several practical applications, while still significant limitations remain to be solved in future work.