Choose a particular American accent sound to focus on, such as "r" or "th."
Watch American television or movies, or focus on American speakers in person if you can. Listen for the sound you chose in the step above and pay attention to the way a speaker moves his mouth when he makes the sound.
Go through a newspaper or magazine article and circle every instance of the sound you chose.
Read the article out loud, taking care to pronounce your chosen sound correctly every time.
Speak with an American, and focus on emulating her speech and movements when you do so. Pay particular attention to the sound you've been practicing.
Repeat this step for other sounds, constantly going back to ones you've already mastered in order to re-seal them in your memory.