In our elementary school, you couldn't request specific teachers for your child. You could, however, request that your child have a teacher with a specific teaching style.
I did request, essentially, not to have a specific 1st grade teacher. (Request letter specified a teaching style that was enthusiastic, energetic, and worked with the kids on different acedemic levels.) The teacher I was avoiding should have retired years ago, and just had no energy left to teach. Plus, her solution for class discipline was to YELL at the kids.
I wish I would have requested a different teacher for 3rd grade. I did not fall into the "rumor trap". I had heard many rumors about how nasty, picky, mean this teacher was. Guess what - this was the worst year of school for my DD. Looking back, and if I was going to do it over, and if I had more kids behind DD, there is NO way I would have them in that teacher's class again.
Luckily, DD ended up in a wonderful 4th grade class, having the best teacher-student relationship you could ask for. Oddly enough though - DD had asked me to request to not have this particular teacher. DD had heard a rumor that this teacher plays "favs" in class. Which was entirely not true.
I don't regret making the request for 1st grade, I wish I would have made a request for 3rd grade. My best advice is - see if your elementary school will let you go in and observe the teachers. You'll learn lots by sitting in the classroom for an hour or so.
Good Luck!