Within the Internet of Things (IoT) and blockchain research, there is a growing interest in decentralizing health monitoring systems, to provide improved privacy to patients, without relying on trusted third parties for handling patients' sensitive health data. With public blockchain deployments being severely limited in their scalability, and inherently having latency in transaction processing, there is room for researching and developing new techniques to leverage the security features of blockchains within healthcare applications. This paper presents a solution for patients to share their biomedical data with their doctors without their data being handled by trusted third party entities. The solution is built on the Ethereum blockchain as a medium for negotiating and record-keeping, along with Tor for delivering data from patients to doctors. To highlight the applicability of the solution in various health monitoring scenarios, we have considered three use-cases, namely cardiac monitoring, sleep apnoea testing, and EEG following epileptic seizures. Following the discussion about the use cases, the paper outlines a security analysis performed on the proposed solution, based on multiple attack scenarios. Finally, the paper presents and discusses a performance evaluation in terms of data delivery time in comparison to existing centralized and decentralized solutions.