There has been increasing interest in connecting disjointed Electronic Medical Records, mobile health data, and related health data systems for the purpose of improving preventative and precision medicine, while also providing individuals with greater access and control to their data. Blockchains provide data transparency, immutability, and decentralized trust -- making them a promising solution to the interoperability and security issues faced by such health data systems. Several papers have proposed the use of blockchain technology in healthcare to determine its viability as a solution and to identify potential applications and challenges. We build upon their work by 1) presenting implementation details related to blockchain applications in health data systems, 2) discussing the security, privacy, and performance trade-offs of each, and 3) identifying a set of research questions regarding the use of blockchain technology in health data systems. We find that blockchain-based healthcare research should place greater emphasis on real-world deployments and testing, smart-contract security, efficient and usable audit tools, blockchain governance, and adherence to healthcare data regulations and standards.