Blockchain as an emerging technology has been gaining in popularity, with more possible applications to utilize the technology in the near future. With the offer of a decentralized, distributed environment without the need for a third trusted party (TTP), blockchains are being used to solve issues in systems that are susceptible to cyberattacks. One possible field that could benefit from blockchains that researchers have been focusing on is healthcare. Current healthcare information systems face several challenges, such as fragmented patient data, centralized systems which are viewed as single points of attacks, and the lack of patient-oriented services. In this paper, we investigate and analyze recent literature related to the use of blockchains to tackle issues found in modern healthcare information systems. This is done to understand issues that researchers commonly focus on, to discover remaining areas of concern in any proposed solution, and to understand the possible directions of the integration of blockchains in healthcare and personalized medicine. Background information regarding blockchains and existing healthcare information systems is reviewed, followed by the methodology used in the preparation of this review, where the research questions to consider are stated. Afterwards, an analysis of the results is provided, concluding with a discussion of the remaining issues that need to be focused on, and how blockchains could benefit the healthcare sector and empower personalized medicine.