Erik Bern did a fun exercise and analyzed the pitch of speakers in various languages: Apparently Dutch is substantially deeper than German (it is also louder, but he did not analyze that).
There was a very definite point when I realized that I had to change my voice to get to the next level with my accent. Oddly enough it was actually while studying German (my third language). It felt awkward at first to alter my voice to the point where I didn’t feel like it was myself talking. But on the other hand I could hear myself sounding so much more German (if you know what I mean). Having been through this transformation I decided to change my “English voice” as well.