Perhaps learning how to mimic accents of people from different regions (Irish, Jordie, Scottish, Alabama, BBC etc etc) of your own mother tongue gives you opportunity to learn how to hear and say different vowels and consonants from another language. Or practice the accent off non-native speakers e.g. a French accent for a Parisian speaking English.
You can normally make assumptions about someone's place or class within a few words or phrases in the UK
EG "fillum" for "film" for people from the North East, "supper" instead of "dinner" for middle/upper class people.
This might be better phrased as a "native accent" for the region you're in. It's another way of saying that others can't pick out anything in the way you speak that identifies you as from somewhere else.
Of course, what is it like to have no accent? All of these people are mentioning Friends, in which they have a "standard" American accent, which says something about American-centric view of the whole debate. I've heard that I don't have an accent (from someone who wasn't American), though I don't know if he meant that I had a vanilla American accent or that I simply didn't have any accent that could be pinpointed.