had this problem with the original version of Civ III straight out of the box. It turned out that I just needed to install the alternate drivers for my ATI graphics card, and that solved the problem.
Then, when I installed the 'morning' version of the patch, the disappearing lines problem (on both the foreign advisor and science advisor) came back.
However, since I installed the 'good' version of the patch, all has been well.
So if you are having problems, try these two things:
1. Make sure you have the correct version of the patch installed.
2. Update all of your graphics drivers.
I am running a laptop: amd-k6 400mhz, 128 mb ram, ATI Rage LT Pro AGP 2x graphics card
There has been discussion in the official "bug fixes" thread here:
http://forums.civfanatics.com/showthread.php?s=&threadid=11671&perpage=20&pagenumber=2
and also there were a bunch of us trying to fix things way back when the game was first released, here:
http://forums.civfanatics.com/showthread.php?
s=&threadid=7613
Hope this helps.