Well, this sucks. GOG should have info like this on the game's main page.
Considering that I'm planning on updating my card anyway, this may cease to be an issue.

So, maybe it's just GOG's version that's glitched? Because I can play it just fine on a 8800GT even with the latest drivers, as mentioned before.

Well, I've managed to fix the problem on my system. I'm running:
Windows Vista 64 Home Premium
Intel Core2 Duo 3.0GHz
Nvidia 8800GT
4GB RAM
I was getting the randome white textures on the walls, floor and ceiling. I did the following though:
Download & Install Nvidia Forceware Drivers 169.02 WHQL
Switch Texture to Max in MDK options (512K)
Switch Render from DirectX+Shiny to DirectX
And now all the randomly missing textures are back in, and the game's running smoothly. To those with Nvidia cards having problems, I highly recommend downloading and installing (after uninstalling your more current drivers) driver set 169.02, or if your card is newer than an 8800GT, then to get the oldest drivers you can. That's apparently what fixed it for me. You can get the old drivers off Nvidia's site by going here:
http://www.nvidia.com/Download/Find.aspx?lang=en-us

That explains things. I only had the software rendered version, that's what I was running. This was before the D3D/3dfx patches were out. And the D3D version does indeed glitch. Haven't tried a Glide wrapper, I'll take your word for that.

GOG support contacted me on Tuesday, asking me to download the game again as they had created a new installer. I tried it out and installed the optional software version. The D3D version has the same glitches but the software version seems to be absolutely perfect. No disappearing textures, no disappearing character, and it runs at a constant 60FPS. I haven't played long enough to know whether some graphical effects are missing from the software renderer, but I can live without them.
If you're still having trouble with this game you should download it again and give the new installer a try. Just watch out - the "Options" button to customize the installation is almost the same colour as the background so it's easy to miss.

Yep, this worked for me too, solved glitch and slowdown problems, just finished first level with no slowdown occurring.

I have an 8600M GT, And I also get this dissapearing texture issue.

Yeah I've been having the same problems, I just play the game on my work computer instead of home.

Hmmm, yeah, I'll have to give the software a shot.
TBH, I think the MDK soundtrack alone is worth $6, but maybe I'm crazy.

I tried to run the MDK_Soft.exe, and it works perfectly fine...and graphic's not that much worse...hell, if you care for graphics, you're wrong here anyways :D

Software mode works for me now too.

So, did they include the 3Dfx version with a Glide wrapper in the updated installer, at least as an option? Good graphics, no glitches - best of both worlds.

Re-downloaded the game to find out - but I'm afraid it's only the software version that's been added, and only as an option within the installer.
Perhaps it has to do with the legal status of Glide Wrappers in general, or the authors of those wrappers more specifically - I do not know.

They don't necessarily need to include a Glide wrapper, then. Just a link or something for those who do wish to get it.
Then again, you can get the 3Dfx executable from the official patch - not sure whether or not that would work with GOG, with the copy protection and all.