KDE and Gnome support the xsession protocol. When you shut down and restart, all your windows are restored to the desktop where you left them. (In Gnome, you have to go to System -> Preferences -> Startup Applications -> Options and turn it on)

From what I've read e17 is supposed to do this too, but I haven't been able to find out where to turn it on or off, at least with Fedora 14. I've burned through half a day and about a hundred links from Google, so I thought I'd ask here - how do I get Enlightenment to remember my desktop state?

I don't think it is possible.
The problem is: i don't care for such a feature. But i just booted e17 (on Debian) and can't find something like that.
It might be it is possible with ecomorph (similar to compiz, but for e17).

The best i can think of:
left-click to open the root-menu -> settings -> settings-panel -> apps -> startup-applications
and
right-click on the window-border of windows -> more -> remember -> advanced -> virtual desktop.

Like said: as i don't care for it it might be i have missed that feature (somewhere under advanced).
Also check the emodules (i had a look, but can find nothing which sounds like that).