I looked around for some export functionality for github but I got a lot of posts complaining it wasn't there. Since I hate applications that take your data and refuse to let you remove it, I was disappointed by this news but further inspection showed that although there might be no "export from github" button, there's an API that more than has it covered. The API returns JSON which is easy to work with from many programming languages, and is perhaps even more powerful and flexible than the simple export I initially expected, so here are some examples.
I decided we only needed our currently open issues, and these were simple to retrieve: http://github.com/api/v2/json/issues/list/joindin/joind.in/open. Since I'm a PHP programmer, I just did this:
$data = json_decode(file_get_contents('http://github.com/api/v2/json/issues/list/joindin/joind.in/open'));
Each of the entries has a comments count too, so for those issues showing a comment count > 1 I grabbed those as well, using http://github.com/api/v2/json/issues/comments/joindin/joind.in/<issue number>
I'll write a post about how we formatted the JIRA import once that has actually been done and shown to work - for now, if you want to get your data out of github, I hope this helps!