First of all (as an American), I have no idea what "standard American" means--there are distinct regionalisms, but no one of them is "standard." While it's probably not disputed that there's an American dialect (usage, spelling, etc.), there's no "standard" American accent.
Off the top of my head, I can think of a number of regional accents (Northeastern, Southern, Midwestern) that can be broken down further (Texas has an accent that is distinct from a deep Southern accent, even though both can be called "Southern"; Boston is known for its non-rhotic "r" sounds, but the same type of non-rhoticity can be found throughout northern New England).
As other answers have stated, there are distinct markers that identify particular regions that, when none are present, the speaker could be said to be speaking "without an accent".