Doesn't Christian mean 'little Christ', that term deriving from the fact that they follow the teachings of Christ, as found in the New Testament Gospels?
I thought the Old Testament was seen by many Christians more as a history of the Jewish nation/race, more often than not highlighting the mistakes they made, so setting out why the new covenant offered through Jesus was necessary.
(Sorry if I've got this wrong, RE lessons were some years ago).
ernie_lynch wrote something not long ago explaining the relationship between the OT and NT very well.