I doubt your child is anxious at all ..it certainly doesn't seem that way from your post have seen this many times in many children and its normal behavior..a lot of children are slow to take to strangers ....
Phew, I agree with pretty much all of the advice above to some extent.
But I wanted to toss in there. Do not, whatever you do, ever refer to her as "shy" because that can become a self-fulfilling prophecy....don't tell people she's "too shy" or "she's just being shy"....try not to use that word at all, because sometimes kids become what they're told they are.
I would say give her some time and try to give her the space to grow individually. Her teacher needs to try to recognize the difficulty she's having and be patient with her, rather than forcing the issue.
At school, is she like this only with the teachers? You mentioned she is loud and active with her peers- does this apply to peers in the classroom as well?
Is the teacher complaining or expressing concern? If she is complaining, I'd be worried about her expectations. If she is expressing concern, you may want to look in to selective mutism, as Rockrose had suggested. I have worked with children with selective mutism who are quite rambunctious at home but do not speak at school.
And one final question... in all of your descriptions, you seem to emphasize that she repeats things and knows memorized answers (saying name falls into this category, as does songs, rhymes, etc). Does she also have independent speech? Such as a conversation that requires an answer that is not memorized? Will she play with toys and have them "talk" to each other or will she pretend with them? (or, not just reenact scenes from favorite shows, but expand on them creatively) If yes, then I wouldn't be too concerned and just give her time. Don't let her teachers pressure her into talking, since it is often due to shyness or anxiety. If her speech is mostly parroting or memorized, then I would suggest you speak more to the teachers about their concerns. If it is due to anxiety or shyness, you can help her by reassuring her in those situations. Hold her hand and give it a gentle reassuring squeeze when in the company of others. Let her know in advance that she does not have to say hello or speak, but that she can smile or wave instead (give non-verbal alternatives). Try not to put her on the spot when in front of others. She is still very young, and pushing too hard now may have a negative impact if this is due to shyness or anxiety.
Hopefully this might help
http://www.parenting.com/article/ask-dr-sears-shy-kid
http://www.babycenter.com/0_shyness_64068.bc
I would say she is shy of talking to other people especially if she doesn't know them maybe the teacher could encourage her more ..maybe teacher doesn't talk to her enough .She sounds pretty bright for her age ..she just needs more outside input and encouragement ...
i would say she's three and shy and to give it time. but if you are seriously concerned about things you could ask your doctor for a referral to a someone like a behavior specialist.
She's still young so maybe she'll grow out of this, but you might want to google "elective mutism". It's an anxiety problem, that makes kids unable to speak in certain situations, often at school, although they're very verbal at home.