Why do Protestants not refer to Mary by the title "Mother of God"? I have heard the title "Mother of God" in connection with the "Hail Mary" prayer that(Ave Maria) is recited by Catholics.  

However, I have never heard this term used in any Protestant setting.  (From the comment by Bobo, we find that the Orthodox also refer to Mary in this way; Theotokos in Greek literally means "Birth-giver of God", as well as Bohoroditza in Russian. Both of these terms are widely used in their respective Orthodox groups). 

Why, then, do Protestants not use this title that appears to be so common in Catholicism?
At its most basic level, Protestants believe Mary merely to be the earthly mother of Jesus - not the eternal mother of "God."  To imply that she had a role in creating that which is divine with Him seems an overreach. This is not to say that Jesus is not God, but Mary's role is as a vehicle for the incarnation, not the means by which the Godhood was made.  To believe otherwise would unnecessarily elevate her above mere mortals.

This is not imply she is not "blessed among women," nor that she was not the vessel through which God incarnated and dwelt among us. But to stress the motherhood of that which is God within Jesus (and no, I do not mean to imply a Nestorian, no-hypostatic union Jesus - he is fully God and fully Man) - that motherhood denotes a role in creation, and that is not possible.

That "Creation" would fly in the face of John 1:2, of course - for Jesus was not created.  Jesus was incarnated, meaning that his existence and personhood indwelt man, but was not created "by the will of man or by the flesh."  Motherhood denotes a much greater role in the creation of the child[1] than incarnation would suggest.

Finally, Calvin suggests that Mary's role as mother was only earthly:


  Opponents of the aforementioned view of Calvin's mariology point out that, in his writings, Calvin never explicitly refers to Mary as the 'Mother of God'. Moreover, Calvin's comments on Mary as the mother of Elizabeth's Lord, may be understood to mean that, in Calvin's view, Mary was mother of the Lord only while he was on earth. Proponents of this view have cited Calvin's commentary on John 19:26, from which it has been argued that Calvin considered the mother-son relationship between Mary and Jesus to have ceased at Jesus’ death. In this scheme, Christ, as he was dying on the cross, appointed his disciple John to take his place as Mary's son, so that he himself might henceforth take his rightful place at the Father's right hand in heaven.


Once John was beheld as "your son," Mary was now the mother of John, not Jesus.



Notes


I was about to say "she does contribute genetic material for example," when I realized that there is nothing to suggest she didn't.  That said, genetics were an unknown science at the time of the incarnation, and as such it is anachronistic to require a knowledge thereof when these titles were being decided.  The point is that Mary did not make God in the same way a mother "makes" her baby.
To be fair, Luther did call her mother of God, but the formulation seems much Roman Catholic as anything.