Like Estate I have some mixed feelings about accent. Partly that I do not have exactly the right accent for where I live, so I'm often not accepted as Australian, even though I am 100% Aussie maayyte! This is terribly frustrating. Usually people thibk I am British or German, which is not a bad thing, but I do not like to lie so it is hard to really capitalize.
However, with voice classes my situation has improved a lot, and I think my Aussie accent comes through more clearly when I breathe and enunciate correctly. But ironically we use stuff like "Four score and seven years ago..." in the voice classes and I discovered I can do a passable American accent if I concentrate. I will be speaking American if I come to US for PU because Aussie accent just sounds strange to American ears. No disrespect but America can be a little inward-looking.
Given that when I'm in Tokyo I try to speak Tokyo-ben and when I'm in Osaka I try to speak Osaka-ben, and that I try to pronounce non-English syllables such as "fu" correctly (sounds like English "fu" but teeth are not involved, only lips, also in Osaka there is less blowing of air through the lips and more glottalization, so it becomes more like "hu")... I see no reason to treat English differently, even though I'm a native speaker. For some reason, it feels fake in English, whereas it does not feel fake in Japanese, but I'm trying to disregard that and become comfortable with adjusting my speech to suit surroundings.
IMO people will be more comfortable with you if you mirror their body language, accent etc. Also you will probably not do it perfectly, so will retain some hint of mystery. Some accents are considered "desirable" though, so it could be an advantage having a posh English accent in US or even an Osakan accent/phrasing in Tokyo, depending on the image you want to portray, but that's just case-by-case IMO.
For Bboy100 I suggest Chase's advice of just inserting a tiny hint of foreignness to make yourself more interesting and gey her wondering, but without overdoing it. I don't really need to follow this advice since I naturally sound foreign and have to work at trying to be "one of us"... so I'm curious to know how it goes.
Ray