I'm an American who lives in Germany and hear many kinds of English spoken by many nationalities.
Just as "one can either write organization or organisation but the main point is to be consistent" I have also generally applied this rule to my speech.
Especially when I was teaching English I would make it a point to not say phrases such as:
even though I thought it was cool. And instead of saying:
Shall we go?
I would say:
Do you want to go?
I can imagine if you are teaching English or if you are representing a company from London or New York or Sydney you have a certain obligation to speak as if you come from that place.
In what contexts is it important to maintain your accent or dialect?