OK, so my eeePC 901 got dropped recently, and the screen broke. It cost me £300 new, and Asus estimated £150 + nearly £60 in shipping to replace the screen. Ouch.
A bit of googling, and I fell onto asusparts.eu, who kindly offered to ship me a brand new replacement screen for just £103. The only snagm obviously, is having to fit it yourself.
Soooo, for anyone in the same position as me (that is, needing to replace the LCD screen on their Asus eeePC laptop), here's the procedure - including pictures!
Firstly, the tools you're need are:
Step 1: Notice your eeePC 901 has a broken screen.

Check!
Step 2a: Expose the screws.

Using the small flathead screendriver, dig out the 6 rubber pads on the screen - there's 4 running across the top, and one in each bottom corner. Don't worry, they come out nice and cleanly with a little pressure. Once you've got them all off, use the philips screwdriver to remove all 6 screws.
Step 2b: You should end up with these:

6 pads and 6 screws - I put them in the order of their placement, just in case they're different in size etc. You can obviously do whatever you want with em! Just bear in mind, you'll be putting them all back in/on later.
Step 3: Separate the casing

This is a little unnerving at first, but you just have to trust it will work. Starting at one of the top corners, wedge the bigger of your flatheads in a few millimetres, and carefully twist to separate the two halves of the casing. As you work around, you'll notice it gets harder to create a gap - this is where the clips are that hold the case together - be careful, apply firm pressure, and they will unclip and separate. Work your way around both sides and the top.
Step 3b: Separating the bottom.

So the same process along the bottom edge - carefully squeeze a screwdriver in, and twist to separate along the entire seam (that you can access).
Step 3c: The last two separation points.

My screen surround was still captive at this point - the last two clips are right by the hinges, and I found it easiest to go in from the front - be careful, using the same twisting technique.
Step 3d: Be careful - there's stuff attached to the surround!

These are only loosely connected, and they can remain attached - they're long enough to let you move the surround out of the way once fully disconnected.
Step 4a: Free the surround from the machine.

The last connections are a clip right at the bottom edge of the hinge it'self. Lean the screen all the way back, and wedge your screwdriver under the hinge as shown. Give it a bit of a jank, it will come free. Repeat on the other side, and your surround should be free.
Step 4b: Move the surround to the side, out of the way.

It can remain connected.
Step 5a: remove the final 2 screws holding the screen in place.

Remove the final 2 screws from the bottom - one on either side, and the screen is free - although still connected by wire!
Step 5b: the screen is free!

Lie the screen down on the keyboard for the next few steps.
Step 5c: Lay down and ready for action.

Messages saying "DON'T TOUCH" just excite me!
Step 6: Carefully peel off the fabric and serial code stickers.

The fabric one especially is very well stuck down, so firmly but carefully is the way forward.
Step 7: Cut the retaining sticky tape.

I missed this at first, and wondered what the hell was going on, but right above the connector itself is a length of clear tape keeping everything held in place - cut this carefully with a sharp knife.
Step 8: Disconnect the plug.

Carefully but firmly disconnect the plug by pulling horizontally away from you.
Step 9: Job Done!.

Well...mostly. That's the screen removed. To replace with the new one, just work backwards through these instructions!