[kafkaesq]

Talk. I wouldn't worry about the "accent" in the sense of vocal register or overall "sound" (which is after all you), but (in the spirit of "don't make people think") it wouldn't hurt to get a precise script of you want to say corrected to be 100% grammatically (and idiomatically) correct, with the help of a native speaker. And to try to get the pronunciation smoothed over to be as correct as you can get it, also. Then practice it enough so that it sounds perfectly natural and not forced (resting for a few days in between, to let it seep in). Basic theatrical technique, in other words.

Because there's something about talking to people in their language on level that's 100% correct (while still not trying to hide the fact that you're a foreign speaker) that somehow really impresses people and makes them want to listen to you. (Which shouldn't be too difficult for you, because going by your writing, your English is really quite good already).