

There is nothing grammatically incorrect about "I'm good."
Good is an adjective. It modifies the subject. The verb "To be" is a linking verb, not an action verb. An adverb there ("well") would be out of place.
"I'm doing well" also works..."well" is the adverb that modifies "doing".
"I am well", as far as I can tell, is grammatically incorrect.
Then again, it's been more than a few years since I've taken an English class, so take it with a grain of salt ;-)

No matter what kind of day you're having, tell them "I couldn't be any better" or "if I were any better it would be illigal" or "a lot better than I deserve." See what kind of response you get to that. I say all three all the time, and it throws people in to an immediately positive mindset about about you.

I've always been surprised and boosted a bit whenever I run into someone who answers "How are you?" with something more enthusiastic, like "Great!" or "Fantastic!" while sounding like they mean it. It's a less common response, so it stands out.

Thought this may help...

UPDATE: It is perfectly grammatically correct to say "I'm doing good." In fact, you should only say "I'm doing well" when you are referring to the state of your health. Thank you, Grammar Girl...
Thanks, everyone, for the comments!
@Steve: You are right. In fact, the link that Hi! left below explains exactly what you wrote. I am a little upset at my high school right now for not teaching grammar (one of those...not one grammar class...but we did spend a month analyzing Hamlet). This should definitely be covered before college.
@Rob: I can see that, but you cannot say that all the time. Would that not sound weird if you say that several times a day or even a week?
@KateNonymous: Yes, I definitely know what you mean. It's great to talk to someone who is energetic, but I guess you cannot really say that ALL the time, can you? Or you should at least not say it and then sound completely fake. And a lot of my how-are-you interactions happen while I'm getting myself that first cup of coffee in the morning, so I feel pretty dead at those times.
@Hi!: Thanks for the link! Very helpful! :-)

(I wanted to post this yesterday, but the image 'captcha' required for posting was not displaying on my browser.)
I'm glad to know that I'm not the only one who goes through this internal monologue with ever conversation. And people wonder why I find small talk so exhausting!