Technorarati's got their Cosmos api, which works fairly well but limits you to noncommercial use and no more than 500 queries a day.
Yahoo's got a Site Explorer InLink Data API, but it defines the task very literally, returning links from sidebar widgets in blogs rather than just links from inside blog content.
Is there any other alternative for tracking who's linking to a given URL (think of the discussion links that run below stories on Techmeme.com)? Or will I have to roll my own?
Well, it's not an API, but if you google (for example): "link:nytimes.com", the search results that come back show inbound links to that site.
I haven't tried to implement what you want yet, but the Google search API almost certainly has that functionality built in.
Is this for links to Urls under your control?
If so, you could whip up something quick that logs entries in the Referrer HTTP header.
If you wanted to do to this for an entire web site without altering application code, you could implement as an ISAPI filter or equivalent for your web server of choice.
Information available publicly from web crawlers is always going to be incomplete and unreliable (not that my solution isn't...).