This may not be completely answerable in a way that applies to all Protestants, but I'm going to put forth an answer that I believe will apply to a large swath of protestant belief.
The generally accepted Protestant understanding of what prayer is can be found at http://www.whatchristianswanttoknow.com/10-prayer-tips-how-to-talk-to-god/
Prayer is simply having a conversation with God. There are many
different thoughts on what you should and shouldn’t say when you pray.
There is merit to many of these ideas. But try to make your prayer
life a simple conversation with your Heavenly Father.
In that sense, prayer is not simply reciting words by memory. That's not a conversation. Prayer is not repeating words that others have said. There are many types of prayer recognized by Protestants:
- Prayers of thanks
- Prayers of repentance
- Prayers of confession and asking for forgiveness
- Petitions (asking God for something)
- Prayers of praise.
In this sense, we recognize that the Psalms and other recorded prayers in the Bible were prayers of the psalmists, or of those that are recorded as speaking them. They are to us a pattern to emulate rather than something to simply be memorized and repeated.
Normally I wouldn't use personal items as part of an answer, but this does illustrate the point, so please forgive me if this strays into the "not a well supported answer territory"...
That said, when talking to people about the definition of repentance, and praying for salvation, in many protestant Churches, we condone a form of recited prayer - we call it the sinner's prayer, and scores of well-meaning Christians when witnessing simply ask the potential convert to repeat a prayer after them, something like "Oh, Lord, I'm a sinner. I realize I'm lost and can't save myself. Please forgive me, come into my heart, and save me", and we call that a prayer.
In a sense, that sinner's prayer is no more a prayer than simply repeating a psalm, and I, personally, like to point instead to psalm 51, and say something like, "Look a this. this is the prayer of someone who knows they're lost, and is truly repentant. Look at how David is begging to be cleansed..." and use it as a pattern for true prayer of repentance.
end of delving into the personal belief portion of the answer
The point of the above journey into not constructive territory is that Protestants, to answer your question, view such prayers as examples - as patterns, but would not see simply rote repetition of the prayers as prayer in and of itself. We would see mere repetition of established prayers as meaningless. To us, it misses the point of a personal relationship with God, because repeating words written by others isn't having a conversation with Him.
Perhaps this article sums it up better, but still says basically the same thing:
How do protestants pray?
Protestants may pray in silence , in groups or outloud. However , they
do not like using formula prayers , and they do not pray to Mary or to