"Three" is often like this. You are going to have to be rock solid firm, mama.
Three meals and two snacks a day. This is perfect for a three year old. Serve them around the same time each day. Water should be always available; reserve milk for mealtimes.
Each meal should have three choices, you pick. No more asking her. (Please believe me, I have lived through this multiple times as a nanny and with my own kid.) Put three reasonable (by reasonable, I mean food she has eaten readily at times in the past) items on the plate, small portions, and that's that. She may make her choice from the plate. Period. If she says "I want such and such" tell her "You have what's on your plate."
Then, be rock solid about this. If she says "I want blueberries" and you have blueberries, then you can say "yes, you can have blueberries with your lunch. Right now, this is what I have for you."
This is what I have for you -- new mantra.
Think of it this way-- at this age, when we ask our kids what they want, you know what happens? They start to think that we don't know how to be in charge of them. It's true. Can you imagine going to work and having your boss ask you 'Well, what do you want to do today?' If it's up to you, you might start something but decide-- maybe that's NOT what I want to do, I'd rather do something else. My boss doesn't seem to know what I should be doing, so why not?
You are the boss. Keep it simple. Expect the first week of this to be potentially tough. You are teaching her that you know what's best for her, so that is going to take some getting used to on her end.
What I often did with kids who had food issues was to get a bigger gladware type container and put a variety of snack items in it for the day. Say, a yogurt, some string cheese, crackers, fruit slices, raisins, lunchmeat rolls--- whatever I knew they would reasonably eat. Then, no discussion-- snacktime was whatever was in the snack box. The snackbox only came out at snacktimes, sitting at the table, and eating together. Do make sure you are sitting down with her, even if just for a cup of tea or coffee for yourself, when she is eating. Kids need the companionship.
I also believe in enjoying my own meal, so if a child asks me for something when I'm seated, I don't just hop up and get it. I encourage them to either help themselves or "I'll get it in a few minutes. Right now I'm eating." This also helps them to remember that you are a person with your own needs and not a waitress.
Be patient-- this usually does happen around this age. I'm not sure why, but I've seen it countless times. Remember, your child will NOT starve to death. Don't become more invested in her eating than she is, okay? And if she doesn't eat her dinner, do as Diane B suggested-- save the plate and re-serve. Don't let her go for the cereal-- many folks fall into this trap with their kids. I think, if they've eaten a full meal, that a bowl of cereal in the evening is okay, but consider that you are setting up the groundwork for how your child perceives meals for the next many years. Good luck!