The original Medal of Honor came out in 1999 on PlayStation 1. Then in early 2002 Allied Assault came out, which I might have played, I'm not sure. The first one I really remember was Frontlines (mid-2002) and then Rising Sun (late-2003).
The original Call of Duty didn't release until 2003, which means Medal of Honor came first. They never quite figured out multiplayer like COD did though and that's why I think the franchise died off.