In my church, the only prayer that we have that is standard and always repeated word for word is our sacrament (communion) prayer. Every other prayer is personalized and individual, no matter the setting. In fact, we seem to be so set against "vain repetitions" that some people seem to think that we have to be excessively creative to avoid even the appearance of repetition. Since the regular members of the church are often asked to prayer in group settings, we are often giving public prayers.
When rote prayer is discussed in church, someone usually brings up other various Christian denominations that apparently only use repeated or memorized prayers. I say apparently because I doubt that there are many devout Christians that never have personal, individualized prayers. When an unreasonable emphasis is placed on avoiding rote prayers, I think there are two (and probably more) problems occur, and I touched on both above.
The first is that we often end up portraying other Christian denominations as having worthless prayers. As I said before, I expect that most Christians say personal prayers like my own that are individual and unique. Our prayers are not inherently better because they are different every time. Rote prayers can be as valuable to the individual, if they are said with real intent.
The other problem is that repetition is not necessarily a bad thing. My prayers are often very similar to each other, but I don't think this falls into the category of "vain repetitions." I always pray for my family, my friends, the prophet, the missionaries, and Jerusalem. I try to have real intent when I pray. I certainly don't always, but I think that the intent is more important than the words.
I do need to make more of my prayers. Prayer is our opportunity to communicate personally with Heavenly Father and I often take it too lightly.