Although it is widely assumed that Artificial Intelligence (AI) will revolutionise healthcare in the near future, considerable progress must yet be made in order to gain the trust of healthcare professionals and patients. Improving AI transparency is a promising avenue for addressing such trust issues. However, transparency still lacks maturation and definitions. We seek to answer what challenges do experts and professionals in computing and healthcare identify concerning transparency of AI in healthcare? Here, we examine AI transparency in healthcare from five angles: interpretability, privacy, security, equity, and intellectual property. We respond to this question based on recent literature discussing the transparency of AI in healthcare and on an international online survey we sent to professionals working in computing and healthcare and potentially within AI. We collected responses from 40 professionals around the world. Overall, the survey results and current state of the art suggest key problems are a generalised lack of information available to the general public, a lack of understanding of transparency aspects covered in this work, and a lack of involvement of all stakeholders in the development of AI systems. We propose a set of recommendations, the implementation of which can enhance the transparency of AI in healthcare.