You have to be an app developer for that to work: /me/threads is accessible only to developers of an app, and there is no permission that you can grant to an app to make it work. Otherwise, you get:

  {
    "error": {
      "message": "(#298) You must be a developer of the application", 
      "type": "OAuthException", 
      "code": 298
    }
  }

You get this even if you try to use the Graph API Explorer tool to view /me/threads.