Speaking with "no accent" is the same like not speaking in a dialect: it means you speak the dialect which has become the "official" one.
Actually, if you record your voice, you will find that your spoken voice has a slight accent, while your "inner voice" (the one you use to compose emails) has no accent. This is because talking to yourself sounds different, due to the acoustics of your head and ears (or whatever).
I think the fact that Socal has become the defacto American accent via TV may be distracting you.
Wikipedia page about it:
It's sad because it's such a beautiful and rich language; the young folks doing the translation are doing it a great disservice diluting it in their rush for volume.
For Arabic; almost every Arab is a gray-beard language nazi! All traces of foreign influence are removed with urgent and great prejudice. Which is good.