I do see what you mean with the breath mints thing, but I can still think of a way to say it without using the term "no offense."
I don't think I have ever used that saying, nor do I think I ever will. I personally hold with Thumper's belief: If you can't say anything nice, don't say anything at all. If someone asks me for their opinion, then the qualifier isn't nessary. If the person hasn't asked, but I still need to say something, I'll think of a polite way of saying it. (IE - Hey, do you want a breath mint? OR Do you have a breath mint I can have? In truth I'll usually use the second, and nine times out of ten the person, if they have one, will pop one in themselves.)
It bugs me though because there are people who are around me who just use it as an exuse to say rude things and get away with it. And if you do happen to take offense (which, a lot of the time we do), they just shrug and are all like "Well, I didn't mean it offensively" as though we're being the immature ones for it. (For an example of how it was used. "No offense, but that's stupid. Why would anyone want to go there?" <-- In reference to a favorite vacation spot.)
So, while you might not mean offense by comment, just becauses you tack that statement on, doesn't mean that you've got a free pass. I really do think people need to think things through before saying them.