Sometimes I find free software on the Internet which does something useful for me, and also comes out clean and green after running it through antivirus and spyware checks, but I still would like to make sure it doesn't send out any data over the Internet to some obscure site.

So the question is:
How can I isolate a specific piece of software running on my Internet-connected PC from being able to connect to the Internet?
I want the software to think it's running on a PC the network cards of which are all down.

Well if you have a firewall installed which i highley recommend, you should just not trust the application you want to block and the program will try to goto the internet but will be blocked.
Something like Zonealarm is pretty easy to use and offers pretty good protection and theres a totally free version, google for Zonealarm Free and you should be able to pick it up.
I have some software on my system here that "trys" to register itself each time I start it up but zonealarm just takes of it and I carry on using the software without any fuss or nag screens. 

Definitely ZoneAlarm. I have been using it for a few years now, and the interface for blocking Internet access is extremely simple. The first time a program tries to access a network (doesn't have to be the Internet) you are asked whether to permit it to do so or no. That way you don't even have to add every single program - you will know as soon as it tries something out of the ordinary