Turns out, whenever you need something on the internet, someone else has already made it.
I was looking around for a way to get several hundred Trello notes (and their descriptions) into a format where I could work on the texts in an offline format. (Trello doesn't have an offline mode.)
I found this excellent extension (made by 'Trapias') which allows you to get your data out of Trello. Click here for the Chrome extension itself and here for the source code over on Github. You can export to HTML, Markdown, Excel and OPML. This is a great set of options, and there are all sorts of advanced selections you can make.
To use:
(1) Go to the board menu -> more -> print and export
(2) a new button will appear called TrelloExport. (Previously there would just have been a JSON export option).