"Are you a french speaker?", I'm tired to ear this question, it always turns my conversations to another subject : from where I come from, my personal story, etc.
I need to get a real English accent so that I'll not be judged only through my French accent.
Perhaps the first thing to do is record yourself speaking. That way, you can listen more closely and hear exactly what is making you sound like a French speaker. Is it the pronunciation? The intonation? The way you phrase a sentence? This will be your starting point.
If you feel brave enough, you could post the recording on Vocaroo, Souncloud or Youtube and put the link in this discussion for feedback and suggestions. I recommend Vocaroo simply because you don't need to have an account there. The recording doesn't need to be long; even just thirty seconds of you reading a text will do.
Your contribution is very helpful, so thank you very much Peachey.