Folder Downloader for Dropbox does exactly what you'd think it would, but lacks all of the style and form of Dropbox's own app. However, if you need an app like this, you'll love it.
In case you couldn't guess from the name, this app does require access to your Dropbox account to work. Giving it the right permissions only takes a few seconds, though. How fast the app actually does its business depends heavily on your connection. Folder Downloader for Dropbox borrows heavily from Dropbox's own layout, but inverts it so there is a lot more white. It can be a pain on the eyes if you're waiting for it to download a particularly large folder. The app crashed once or twice during testing, but that problem seemed to be on Dropbox's end, and not a fault of the app, itself.
There are a few hiccups, but the app gets the job done in most cases. It surely won't win any design awards, but Folder Downloader for Dropbox will let you get all of your photos and documents off of Dropbox. If that's all you need to do, you won't have any reason to complain.