You might be suprised at how little most P3s consume-

http://saf.bio.caltech.edu/saving_power.html#rawbox

http://en.wikipedia.org/wiki/List_of...on#Pentium_III

Apart from a few of the dodgy models (e.g. P3-600 'katamai') the P3s are under 35watts TDP, which is pretty cool compared to most 'normal' desktop CPUs

You can passively cool P3s as well, if you dont mind downclocking a little or have access to a nice big 'socket a' cooler.

Atoms can be 'low power' but they arent all like that. The original atoms came with dodgy interl chipsets that chewed though far more power than the CPU (early Atoms with a cooling fan normally have the fan over the chipset, not the CPU....as the chipset is far hotter than the CPU is (22watts TDP chipset IIRC))

Hardly. Basing that from your guess about P3 power consumption are you?

Sure, newer computers will run faster...but they can also use more power than the old P3 systems (and apart from intel atom/AMD fusion systems, newer system will eat more power even at idle than the P3).

AFAIK cmov is an x86 instruction that only appeared with i686 CPUs.

If you try to run an i686 compiled disro on a i586/i486/i386 class CPU this is one of the issues you can hit.

Try finding a distro that isnt compiled for i686, probably an i586 distro.

Theoretically you can passively cool anything if you have a big enough chunk of copper. My Atom and chipset were passively cooled by an approximately 3x7 inch sheet of aluminum that is under 1/8 inch thick.

Mine is one of the Intel 945GSE based ones, the chipset is 6W TDP, the processor is 2.5W TDP. According to powertop, my whole system is using less than 10W while browsing with a few things in the background, which may not be totally accurate, but since my battery monitor is fairly accurate, I'd assume it is close enough. I can get it down to 8W if I dim the screen. That is including a wireless chipset too. Streaming a 1080p video from my file server maxes it out at 13W.

I guess I was a little off on my estimate of a PIII, I remember my 1ghz PIII used to run really hot.

I can confirm that Slackware 13.37 (latest stable release) runs on i586 CPUs if you use the single processor kernel. (Slackware is compiled with CONFIG_X86_MINIMUM_CPU_FAMILY=5 and optimised for i686 CONFIG_M686=y). The installer will require >64MB of RAM to boot successfully.
The above machine has 128MB RAM and will run a light window manager (I use Windowmaker). You would likely need a strong bladder as you can drink a lot of coffee waiting for things to happen.