I don't expect any serious,well defined,oscar winning stuff with Godzilla-this is a giant monster movie with the basic premise (if you watch the good ones,and in the original Japanese) that it could be
the end of mankind if we don't get off our collective butts and find a way to stop Godzilla and all the other monsters that show up.
Cloverfield has been mentioned,while that was a pretty decent film but if you want to have the real tone of what Godzilla is think of him in New York...and then think of a NYC that looks like a atomic bomb went off,thousands of dead,thousands more injured,the military suffering major casualties fighting back,the ones who go in later knowing they're probably gonna die-THAT is Godzilla.
And frankly I know this may,may sound corny but I think they should use a bit of the original music in the new American version,in particular the stuff when the military goes diving in is classic and no reason it won't work on the new film.
THIS music:
Gojira's (Godzilla) Theme Song - YouTube
I feel in some ways you need some aspects of the old films in any remake not just Godzilla though,things like music,and the "feeling" needs to be there too.
That's my two cents,for what it's worth.