Since version 2.6.21, the Linux kernel has had a feature called tickless. This was implemented to help reduce the power consumption by keeping the CPU in low power mode for longer periods. Ever since I started using Linux I have been under the impression that because it's less bloated than Windows it would be much less power hungry. However from what I've heard and seen there doesn't seem to be any big differences. I have even heard people say that XP is less power hungry than Ubuntu for example.
I have never tried Linux on a laptop so I don't have any experience of this. Phoronix did a test last October to see the power consumption of XP, Vista, Fedora 8 and Ubuntu 7.10, you can see the results here. It doesn't look very good for Linux. I'm sure there are some very lean distros out there that use very little power but they tend not to be very user friendly. All this however is not going to deter me from installing Ubuntu on the eee pc 1000.
To help Linux users reduce their power usage, a handy little tool called PowerTop has been created. This application will show you the software components that are preventing optimal usage of your hardware and make suggestions for both hardware and software configurations to reduce power consumption. You can get a hold of this program through synaptic. Once installed open the terminal and type in sudo powertop. I have included screenshots of the program with some power saving suggestions.
LessWatts.org - PowerTop