We are using Trello and found it rather useful. Is there any way to have an RSS reader connect to our Trello account and show us all our cards?
No there is no way to currently do this. It is in the archives as a feature request.
I'm not sure if/when Trello will ever implement RSS feeds, so I went ahead and made a small web app that will generate them for you. It should work with either public or private boards (if you give it an auth token).