There are a lot of (open source) projects that I'm interested in - dozens at least. I'd like to get a notification whenever a new version is released.
Is there a service that does this?
Freshmeat is a site that lists new releases of open source projects continually. If you register, you can subscribe to projects and they e-mail you whenever a new release is there.
If the project you want to follow isn't listed there, you can always see if you can follow them at the hosting site the project uses, since they increasingly allow these social features (typically inspired by GitHub's follow functionality).
And ofcourse, lots of projects have a dedicated Twitter account to report on new releases.
Touchpine is a service which allows users to get daily email notifications about updates for software they are interested in. If the software is not already listed you can request it and in most cases it will be added within few days.