Everything installed okay, but when I try to run the game the computer tries very hard sending me to various black screens as it tries to get the thing to run... then it gives up. Do I need to do something particular to get the game to run?

I'm running Vista Home Premium.

might be a problem with dosbox on vista.
dosbox vista wiki

I don't think so, my other DOS games work fine with DOS Box (well the traditional way of mounting the folder I want).

Although I did get it to run by moving the game to my default dos games folder (it's called 'oldgames' , guess what tutorial I read to get dos box running?) and mounting the folder... oh well, if that's how I have to run it, that's how I have to run it.

hahah, I used the same tutorial to play the demo.

I figured it out.

You need to go to Graphic mode setup (you can find it on your start menu under GOG.com->Redneck Rampage Collection) and set it to either "Overlay" or "OpenGL". The program seems to default to "Surface" which doesn't work. You'll have to do that for all three games.

It doesn't run perfectly (graphical glitchs start popping up when you bring up the menu) but at least I can access the program now.

My setup is:
Vista x64
Intel Core 2 Duo CPU E8400 @ 3.00GHz
NVIDIA GeForce 8600 GT 512MBs
3GB of RAM

Right click on the short cut and select run as administrator. The game should then run.