I think I have a keen sense of awareness when I am speaking. I listen to how words come out of my mouth. Even if you do not feel you can hear your own accent when speaking, you should be able to hear it if you record yourself and play it back. Listen to a recording of a native speaker of your target language. Listen to just one sentence. Concentrate on it. Get an audio editor like Audacity and save just that one sentence to a file. Convert it to an MP3 and then import it into iTunes. Then play that file in a loop. Listen to it 25 times at least. Then begin practicing the sentence. Say the sentence at the same time as the recording. Match your speed, intonation and rhythm with the recording. When you think you've got it, note how many times it took for you to get it. Was it another 25 times? If so, continue repeating 25 more times along with the recording. However many times it took for you to "get it" is the number of times you should continue. Then you will have doubled the number of times you have said this sentence. Think you're perfect now? Then record yourself and listen to your recording. Now what do you think? Do you sound like the native speaker you were listening to? If not, how do you differ? Where do you need to concentrate more on? If you follow this kind of training for a while, I believe you will develop a keen sense of awareness. You will be able to hear the way you speak even while you are speaking. From this point, you'll be able to continually improve your accent in the language. It won't take long before people mistake you for a native speaker!
EDIT: I forgot to mention, after you try out this suggestion, come back here and give me a report. I would love to hear how it went for you.