http://bugs.winehq.org/show_bug.cgi?id=6351
--- Comment #89 from Mark I. pilota51@gmail.com 2011-10-19 10:43:29 CDT --- Sorry, it turns out they are completely different glitches, at least as far as the workarounds.
The "AlwaysOffscreen" workaround does work for the black lines all over the ground (as seen in Adam's screenshots), as does enabling AA.
To get rid of the flickering black hexagonal patches seen at mid-long distances (mostly visible in Robert's last screenshot), I had to set the terrain quality to low.