Cheers!

[Grace]

"if Americans don't normally say "Cheers!", I wonder why that name was given to the bar in the sitcom. Were the producers aiming at an "exotic" effect?"

I have heard many people say cheers here in America. It's likely it's more of an American thing than British, though I don't know whether the show started it or not... 

But I've been to or hosted many a dinner party where someone in the group would say cheers.. I don't personally say it but I've heard it enough to see it as fairly common, especially among those who aren't religious..

Do you mean you've heard it at a meal where religious people would have been saying grace? That's an interesting angle.

Yes...  and that's always how it seemed to me.