Yes .. but I think your idea of Downs is a bit rugged for somewhere so close to the Shire perhaps a wilder part of middle earth... There are barrows on downs on the Isle of Wight ( a short ferry ride away from me) . The last Wight walking festival had one event that was "Walk the Barrow Downs"!!!
http://www.megalithic.co.uk/article.php?sid=7055