While researching this topic, I noticed that all the examples given appear to be for the Dropbox v1 API, which has been deprecated and will be turned off in a few months (June 2017):
https://blogs.dropbox.com/developers/2016/06/api-v1-deprecated/
The v2 API looks much more complex - there doesnt appear to be any direct equivalent for the method suggested above (single HTTPS call with an API Key). The authentication requirements now seem more complex:
https://www.dropbox.com/developers/reference/auth-types#user
It's stumped me for now - has anyone got this working?