One possibility would be to make a JQL query, save it as a filter and then subscribe to this filter.
1. JQL query
The following query gets all issues that are assigned to me and had a dependency resolved in the last 24 hours:
assignee = currentUser() and (
issueFunction in linkedIssuesOf("resolutiondate >=-24h", "is necessary for")
issueFunction in linkedIssuesOf("resolutiondate >=-24h", "blocks")
And this is the reverse query, it gets the corresponding issues that have been resolved and are therefor no longer blocking my work:
resolutiondate >=-24h and (
issueFunction in linkedIssuesOf("assignee = currentUser()", "is dependent on")
issueFunction in linkedIssuesOf("assignee = currentUser()", "is blocked by")
2. Save as Filter
Copy one of the above queries into the search, hit enter to see if everything works, then press "Save as", name the filter and submit.
3. Subscribe to Results
After saving the filter, the "Details" link becomes available and there you can set up a subscription. For the above filter a daily e-mail makes sense. You can adjust the period, e.g. to a week, but make sure to keep the filter and subscription settings in sync. Otherwise you will either receive the same issue in various mails or miss some issues completely.
I found this solution while researching my question. If there is an easier way I'd like to learn about it!