I find it's entirely possible to speak without an accent when one tries. When in the company of intelligent people I tend to speak quite eloquently, in fact. Whenever I'm annoyed or simply in the company of fools that tendency vanishes, though, and I'm back to my good old somewhat Irish accent.
In all of the universe there is only one person with your exact charateristics. Just like there is only one person with everybody else's characteristics. Effectively, your uniqueness makes you pretty average.