It's official: application/vnd.geo+json is the media type for GeoJSON registered with the IANA. I had been holding out for a profile of application/json, but it seems that the internet has resigned itself to a Cambrian explosion of JSON media types.
Start serving your GeoJSON up with the new content type
HTTP/1.1 200 OK Content-Type: application/vnd.geo+json
and start asking for it by name.
Accept: application/vnd.geo+json; q=1.0, application/json; q=0.5
Update (2019-11-13): Since the publication of RFC 7946 GeoJSON has a new official media type, application/geo+json.