Back at school I had a French teacher who wasn’t a native French speaker, but still spoke the language fluently. His grammar was perfect, he had a wide vocabulary, but many students just could not take him seriously. Why? Well, he spoke French in what can only be described as a broad British accent. While a French person wouldn’t have much difficulty in understanding what he was saying, he never at any point sounded like a French speaker.
However, it is important when learning another language to learn the accent along with it. As this article says, an accent will help you in many ways – namely, improving your clarity, as well as fostering intimacy and promoting respect.
As regular readers will know, I’m currently learning Mandarin Chinese, and am in contact with others who are also learning the language. Many native American English speakers I know who learn Chinese make very little effort to affect a Mandarin accent when they speak, which is a much larger problem with Mandarin since it is a tonal language, and when speaking a tonal language in an atonal accent, it’s much harder to distinguish what tones the speaker is using.
Likewise, I know many foreigners who have learned English, and while many of them have around the same level of fluency, some have worked on their English accent so much that they are almost indistinguishable from native English speakers, while others still sound like a foreigner speaking English (albeit very well!).
So, don’t be afraid to emphasize your target language’s accent when speaking it. Of course, conversing with native speakers (or perhaps watching movies and TV shows in the language) will help you master the intricacies of the accent.