It's true in a general way in the U.S., but not nearly as true as some might think.

There are some areas, both big and small, that have distinctive regional accents.

The biggest difference of all is between North and South, with the Far West possibly becoming more distinguishable than it used to be. The Southeastern and Gulf Coast is also distinctive. New York City and New Orleans can be surprisingly similar.

It's complicated further by various ethnic accents.

It's often difficult for most Americans to make more than a fair guess as to where a stranger might be from. It often takes a professional dialectologist to identify an accent any more precisely.   

Britain has more and more distinctive regional accents in a far smaller area.