It depends on the word. I was a broadcast journalist for a few years and they had very specific guidelines for pronounciation.
It wasn't horrible for me. I grew up in Chicago and didn't have much of an accent. My friend from TN had to take tapes with her and practice pronouncing different words. This was years ago but one that I remember she had trouble with was inny (like a belly button) and any. They sounded the same when she said them.
Oh. I forgot to mention that this now haunts me because I work near Boston. Alarms go off in my head when I hear the locals talk. :)