Please open your debug.err file in notepad, and link the contents into a post here via pastebin. This will give us some technical information that can help troubleshoot your problem.

The file can be found in My Documents\My Games\Elemental

Be sure to copy the debug immediately after the error occurs; if you start the game again, the debug log will be erased and recreated for the new session.

(Do note that the x800 is below the system requirements)

I don't know if this helps - but I had very similar graphics problems with a different game (and my PC is waaay up there as far as requirements go). Turned out that my graphics card overheated - you might want to open your PC and see if it's too full of dust - after all, PCs typically dust themselves full...

Oh, and dare I suggest a more powerful PC - you can get rather nice systems second hand on eBay... 

I have pretty much the same system as the original poster (Windows XP Home, Pentium IV 3.2Ghz, Radeon X800XT, 1Gb RAM), and have experienced the same exact issues since updating to 1.07. Unlike him, I haven't narrowed the issue down to any specific in-game occurence, as it happened once right after new world loading, and another time after I started to build a farm on fertile land. It's hard to describe, but basically some artifacts appear, looking somewhat like the first screenshot of the OP, but they look differently at different times. The first time, on world loading, it looked white, and the second time, green/gray. The artifact seems to stretch from the gameworld ground upwards into the sky, and appears to be some kind of a 3D grid-type thing (excuse my limited game engine knowledge). This might be completely off, but it seemed to me that it's shape looked somewhat like what a ray of sunlight would look like coming down, if instead of being clear and yellowish, it was opaque and those colors, but that could be completely off. If I get it again, I'll try to link the debug file.

My computer sits on my desktop, and it has a clear side panel with all sorts of blue LED components inside.  I blow the dust out on a regular basis so that I'm not embarrassed to have someone look at it 

While I don't have a temperature probe on my graphics card, I do have one on both the motherboard and the CPU.  At idle, the temperatures are 100°F on the CPU and 95°F on the motherboard.  While playing, the CPU climbs to about 135°F, and the motherboard stays the same.  Max temperature for my chip is 152°F, so I'm well below that.

I have done some experimentation, and have found that putting the mip map levels to 8 instead of default reduces the occurrence of these artifacts.  None of the other settings that can be found in-game had any impact on the glitches.  I did find that modifying the "EnvironmentProps" variable in the Prefs.ini file would change the glitch behavior.  The default setting of "4" (at least it was default for me) produced the best results, while any number higher or lower made things worse. (Hilariously worse in the case of larger numbers.  While the "explosions" still happened, the ground was also covered in lines looking something like a giant thumb-print)

I also tested various screen resolutions, and windowed vs. full screen, and it didn't seem to make any difference.  I "think" that things were better when windowed using a non-widescreen resolution, but I don't have any evidence for that assumption.

I have also noticed a tendency for certain meshes to cause these glitches.  The most prolific are the "old growth forests".  These tiles would glitch most frequently (9 times out of 10), with the Empire hovels second (6 of 10), and ancient battle grounds third (4 of 10).  Being "exposed" (in line-of-sight) or in fog-of-war didn't change anything.  Both locations were equally likely to glitch.

I know that there isn't much that can be done, with my card not being supported and all, but maybe someone else can use this info and figure out which graphic frill is causing this and figure out how to turn it off 

Grabbed the latest drivers, didn't help at all. Same exact thing:

http://pastebin.com/SV9rT5bN

On top of that, this graphical corruption seems to be getting worse with either more turns or more huts and adventure objects on the map, those glitches are now all over the map and restarting the game doesnt help:

Just wanted to post an update here for anyone experiencing this kind of issue. Kryo posted a useful suggestion in another thread, so i just want to re-post his suggestion here, since it helped me get rid of the above problems. Basically, just make sure your video card drivers are not installed over older ones. Uninstall them completely, possibly using something like Driver Sweeper to clean left-over files, and then do a clean install of the latest drivers. Once I did that, the graphical glitches went away.

I just wanted to say that I have this same problem on my older desktop PC.  What I experience can be best described as someone stretching my resource nodes through a taffy machine.  

Again, I'm using the ATI X850XT GPU.  I don't think this is an overheating issue if we're all getting the same thing, it's obviously limited to this card.  It's odd that this card is "below" the system requirements.  Even though the X1600 came out later, the X800 series is the more powerful card.  The X800 benchmarked much closer to the X1800 (minus the HDR capabilities).  So I don't think it would be a lot to ask for this issue to be fixed from Stardock.

In the mean time, if anyone has a silver bullet fix for this issue, I'd be very happy.  I'd like to purchase a second copy of the game for my older machine for multi-player, but I wont do that until this is resolved.

Yeah, I can understand that.  Problem is that 7 years ago I can't imagine many people, in their right mind, going out and buying an x1600 card when there was the X800/X850 card that performed at a higher level for about the same cost.  The X800 series was extremely popular back then.

Anyway, if that's the stance, I'm okay with it.  I suppose it's nitpicking to want a game in 2010 to run perfectly with a card from 2003.  Hopefully there is a tweak or setting causing it that someone can find that can at least ease the pain a bit.  I know that Second Life had this very same problem when I played it a few years ago.  Googling the issue revealed that by simply turning off a video element would fix it.