Godzilla, yes, but also Bewitched, Dukes of Hazzard, SWAT, House of Wax, Planet of the Apes, War of the Worlds, Texas Chainsaw Massacre, The Avengers, Psycho (Vince Vaughan?!).... the list goes on.
The only exceptions to this rule:
1) The Canadian remake of "Dawn of the Dead", starring Sarah Polley and Ving Rhames. The director (Zac somebody, of course) had the good sense to no even attempt to copy Romero, and instead took the movie in an entirely different direction. (Maddox' review says more than I ever could:
http://www.thebestpageintheuniverse.net/c.cgi?u=dawn_rules2) John Carpenter's "The Thing". Ever see the original? Carpenter took the original, which was basically just another 50's creature feature and turned into an incredible, tension-filled epic of paranoia and really, really cool monster effects.
All other remakes can kiss my butt.