in my experience running 720p on a Pentium IV in XP, is like running a marathon without legs over miles of broken glass as your being mauled to death
by a ravenous Bengal tiger.. (previous pc)

my current computer is a Mac Mini - Core 2 Duo processor 2.0 ghz, with 4 gigs ram, attached via firewire 800 to an external Lacie terrabyte drive.
i boot from that drive obviously.
runs 720p silky smooth in vlc, quicktime etc.
for 1080p i use Plex which runs smoothly as well.

to all those claiming those crappy net-books are smoothly running 720p; i love to see a screen-shot of an intense series such as Evangelion 1.11
You Are (Not) Alone with a cpu index.
me running it 1080p.
sure there are dropped frames but it is
silky smooth and the cpu(s) are not being strangled. here it's 43%. fairly respectable for 1080p.
net-books are almost avant garde in their own silly little way. low technology becoming the "it thing" with bemused dipshit tech types.

so in answer to your query- at a minimum Core 2 Duo processor.