HaD MMU question reply

Wow, Hackaday seemed to have not submitted my rather lengthy reply, aye.

Like Artenz said, you should go for a chip which includes an MMU.

The gist of it was there is an effort named uClinux a few years ago which let you run an MMU-less version of the kernel, and was recently mainlined into mainline Linux. Here are a few links which might help you out for finding out the implications of running MMU-less is.

http://free-electrons.com/docs/uclinux/
http://electronicdesign.com/embedded/practical-advice-running-uclinux-cortex-m3m4
http://www.alterawiki.com/wiki/UClinux

You can also just emulate an MMU based system on an AVR! Though it is slow as hell, like two hours to boot slow.
http://dangerousprototypes.com/2012/03/29/running-linux-on-a-8bit-avr/