My desktop is a Windows 7 64 bit with 3072 mb RAM, DirectX11 support, and I have a NVIDIA GeForce 7800 GTX video card (a used one without the fan, making it run less than optimally). The question is, can I run it? [to put things into perspective, I can run Crysis smoothly, but on the absolute lowest settings (i.e. parts of the environment, namely some rocks and trees, spawn when I'm close enough).

Do you have 256MB or 512MB version of the card?

Your card has shader model 3.0 support, and if you have 512MB version of the card, you should be fine.

Somebody has tested the techdemo with 256MB of video memory and said it was running smoothly, but official minimum for video memory is 512MB.

Also, dualcore CPU is stated as requirement. What cpu you have?

If you want to make sure you can run the game, you could download the compability test they did a while back.

You can no longer download it from the main page anymore, but can be found here:
http://www.atomicgamer.com/files/95753/legend-of-grimrock-hardware-test-v1-0-4#

It's a 397 mb download, you have no game play, but you can pan the camera around a bit.

I used the compatibility test, and I got an error message saying "D3D Error - CreateTexture failed: D3DERR_INVALIDCALL" (My drivers are updated, and I have DirectX 11 support).

There is discussion in game's official forums about this, and the developer says it is fixed:

http://www.grimrock.net/forum/viewtopic.php?f=12&t=207

No you don't. Your OS supports DirectX 11, your card doesn't.

You have a DX9 card, it can't even run DX10.

What I recommend is this:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814125397

This card will keep you going for another 4 years or so.

If you have an Intel Core i3 or better CPU, you could disable the video card and run it on the on-chip GPU.

I got my version working and it doesn't have a good graphic card though...a intel gma x4500 and this isn't my gaming rig o.o;;;