This term comes form the old Anglo-Saxon Rod, or Rode, and means, simply, "cross."
Originally the Rood referred only to the True Cross itself, the one on which Jesus was crucified (this explains why some ancient churches, particularly in the British Isles, are dedicated to the True Cross, or Holy Cross, and are named Holyrood, of which the Scottish Abbey and Royal Palace are perhaps the most noteworthy examples.)
In the medieval church, the custom of having a representation of the true cross, within the church itself, developed.
As a result, these crosses, or roods, were typically quite large, in order that they could be seen from all parts of the church, and were placed on a gallery, screen, or, as in the case of our rood, hanging from a beam spanning the chancel arch.
By the 14th century almost every church in Western Christendom was appointed with a Rood.
The original design of most roods was fairly austere: a mere representation of the crucified Christ. However, by the 15th century, many roods were richly carved, painted or gilded, and with the arms of the cross terminating either in fleurs-de-lis, a symbol of the Holy Trinity, or in emblazoned medallions of the symbols of the four evangelists.
What is interesting to note is that, despite the controversy over its design, our rood is fairly traditional.
While the crucified Christ is not actually depicted, one could argue that He is symbolically represented by the crown of thorns that is placed at the cross's apex; note also the traditional inclusion of the fleurs-de-lis at the rood's base.
As far as whether or not a rood should include an image of the crucified Christ, there is no right or wrong answer.
Since the Reformation, crosses in general have been typically unadorned, not only proclaiming that Christ is Risen, but also emphasizing that we are an Easter people.
However, it is not inappropriate for crosses to have an image of the crucified Christ since, as those of us who make our Easter pilgrimage with Jesus know, the road to the Resurrection passes through Golgotha too.