Current telehealth services are dominated by conventional 2D video conferencing systems, which are limited in their capabilities in providing a satisfactory communication experience due to the lack of realism. The "immersiveness" provided by 3D technologies has the potential to promote telehealth services to a wider range of applications. However, conventional stereoscopic 3D technologies are deficient in many aspects, including low resolution and the requirement for complicated multi-camera setup and calibration, and special glasses. The advent of light field (LF) photography enables us to record light rays in a single shot and provide glasses-free 3D display with continuous motion parallax in a wide viewing zone, which is ideally suited for 3D telehealth applications. As far as our literature review suggests, there have been no reports of 3D telemedicine systems using LF technology. In this paper, we propose a cross-platform solution for a LF-based 3D telemedicine system. Firstly, a novel system architecture based on LF technology is established, which is able to capture the LF of a patient, and provide an immersive 3D display at the doctor site. For 3D modeling, we further propose an algorithm which is able to convert the captured LF to a 3D model with a high level of detail. For the software implementation on different platforms (i.e., desktop, web-based and mobile phone platforms), a cross-platform solution is proposed. Demo applications have been developed for 2D/3D video conferencing, 3D model display and edit, blood pressure and heart rate monitoring, and patient data viewing functions. The demo software can be extended to multi-discipline telehealth applications, such as tele-dentistry, tele-wound and tele-psychiatry. The proposed 3D telemedicine solution has the potential to revolutionize next-generation telemedicine technologies by providing a high quality immersive tele-consultation experience.