Why and when did allegorical interpretations of Scripture lose their appeal? Gregory of Nyssa's book The Life of Moses is, if nothing else, a very interesting read. He goes through Moses' account in Exodus and ascribes spiritual meanings to every little detail, although it is doubtful that any of them would pass muster in a pulpit today.
Augustine would also do allegorical interpretations, such as the following:

“Thus as great and brilliant a scholar as Augustine offers the following interpretation of the parable of the Good Samaritan:
A certain man went down from Jerusalem to Jericho = Adam
Jerusalem = the heavenly city of peace from which Adam fell
Jericho = the moon, and thereby signifies Adam’s mortality
Thieves = the devil and his angels
Stripped him = namely of his immortality
Beat him = by persuading him to sin
And left him half-dead = as a man he lives, but he died spiritually, therefore he is half-dead
The priest and the Levite = the priesthood and ministry of the Old Testament
The Samaritan = is said to mean ‘Guardian,’ and therefore means Christ himself
Bound his wounds = means binding therestraint of sin
Oil = comfort of Good hope
Wine = exhortation to work with a fervent spirit
Beast = the flesh of Christ’s incarnation
Inn = the church
The morrow = after the Resurrection
Two-pence = promise of this life and the life to come
Innkeeper = Paul

The above interpretation, however, is a requote from Gordon Fee and Douglas Stuart's classic and authoritative book How to Read the Bible for all its Worth, as an example of how modern interpretation is NOT to be done.  They are right, of course, because allegory really isn't testable in the end.
My question is this - how did the vast majority of modern Christians come to realize that allegory was not the best hermeneutical principle for reading Scripture?  Historically, when did the shift occur, what triggered it, and how did Chalcedonian Christianity come to realize this was not the way to read the Bible?
Along these lines, are there any major strains of Christianity that would accept this? (My guess would be that if anyone still does this, it would be the Eastern Orthodox, but I'd need confirmation)
I'm not sure it completely has abandoned allegorical interpretation.  The PCA church still takes much of Revelation to be allegorical, although the rest of the United States doesn't so much anymore... a result of John Nelson Darby and Cyrus Scofield in the 19th century, who came out with the Scofield Study Bible, with Darby's dispensational, and somewhat more literal interpretation of Revelation. 

Many of Jesus parables are still taken allegorically, though they're often coupled with a philosophical Occam's Razor in order to come up with the simplest explanation.  

I don't think it's that we don't take parts of the Bible allegorically, I think it's just that we no longer go out of our way to do so, unless the text seems to call for it. 

That being said, much of the shift that you're referring to, at least according to Wikipedia, comes from Luther and Calvin, the former of whom came up with the principle of scriptura sui ipsius interpres (Scripture is it's own interpreter), and the latter of whom came with the principle: brevitas et facilitas, brief and relevant.  That, combined with the rationalist and humanist movements of the 15th and 16th centuries combined to make the shift.