Was the mode of execution for Jesus required to be crucifixion? Did Jesus need to die on a cross or were there alternative endings?  Would being stabbed with a sword or if he died during the beatings have been good enough?

Specifically, are there prophecies held by various denominations that require the Jesus' method of execution specifically to be crucifixion, or was it simply enough that Jesus die in some fashion?
No Direct Theological Significance

Theologically speaking, crucifixion had no theological significance in itself.  Jesus had to die for sins, and God, in His omniscience, foresaw the manner in which this would happen.  Then He inspired prophets to predict that kind of death centuries before it happened.

However, it did, perhaps, have some symbolism.  

The Tree of Life

In the Garden of Eden, there were two trees--the tree of the knowledge of good and evil and the tree of life.  Adam and Eve ate from the tree of the knowledge of good and evil, and sin entered into the world.

We don't know what happened to the tree of life, though.  Presumably, if they had eaten of it, they would have lived forever in their separation from God.

Four thousand years later, though, another tree is introduced--the Cross of Christ.  Jesus takes our curse by being "hung on a tree".  In essence, then, the Cross is the new Tree of Life.  Anyone who receives what hung on that tree receives life.

Interestingly enough, if you take fruit from a tree and remove the seeds, then you bury the seeds, then new life comes.  Jesus was taken down from Calvary's tree, buried, and then came forth with new life.

Conclusion

So, crucifixion was not theologically necessary per se.  However, it does fulfill prophecies that God gave us, knowing how Jesus would die.  It also seems to be symbolic.