One thing I found can be helpful is to imitate a native speaker, as if you are doing an impression of them. I think there is a part of us which remaps phenomes from a foreign language in to our mother tongue, which creates the accent. However, if you imagine you are mimicing someone, like a comedian, a different mental process happens and you replicate more of the nuances.

When I was trying to learn Mandarin I had the good fortune to get instruction from a friend who is a native speaker and a concert pianist.

In order to teach the tones (chinese uses pitch to make vocabulary distinctions) she convinced me to sing the words. She literally wrote the tones out on staff paper (i am also a musician) and i treated the phrases as little songs. It worked surprisingly well.

I realized that it was really hard to make myself speak that way. I felt ridiculous singing at people. In a certain sense it felt artificial, as if I was making fun of them. But I think that is what it takes to start the process of speaking new sounds. You have to let go of "your" voice and "your" way of speaking, which is rooted in your native language's sound, and almost assume a character who sounds like a different person.

Full disclosure: I don't remember much Mandarin anymore. But if i were learning Spanish I would try the same thing. I would imagine myself as a Spanish dude and "put on a Spanish accent" as if I were acting in a movie.