All credit goes to Rusty!, ChainsDD and Koush.
Step 1: (Download software, boot phone in APX-mode).
First you need to download Rusty!'s excellent ClockworkMod NVFlash-installer. (Mirror.) Then download the SU303.zip-package attached in this post and place it on your external SD-card, if you have one, if not place it on the internal one. Extract the contents of Rysty's .rar to a directory on any drive, I'll refer to this as C:\NV, but you can name it anything you like. Power off your phone and remove the battery. If you are using a desktop computer, plug the full-size USB-plug directly into a port on the motherboard (backside of the case usually). Press and hold volume-up AND volume-down while plugging the micro-USB into the phone.
The next step is for people (well, computers really) who have not used NVFlash before. If you have, skip to Step 3.
Step 2: (Install APX-driver.)
If you have not used this mode before, called APX- or NVFlash-mode, your computer will indicate it has found a new device it has no driver for. The screen on the phone will remain black.
Right-click My Computer, select Manage, then Device-management. You should see a device with a yellow exclamation mark. Right-click it, select update/install driver. Select to manually search for a driver. Browse to C:\NV\driver, click Next. You should be greeted by a message telling you that «NVIDIA USB Boot-recovery driver for Mobile devices» has been installed.
Step 3: (NVFlash ClockworkMod.)
Navigate to the C:\NV folder, double-click the script called «flash-recovery». If you have an external SD-card press 2, if not press 1.
If you have an SD-card, double-click the "install_external_xp" batch-file, if not double-click the "install_internal_xp" batch-file.
The phone will display a Software-upgrade-screen, while the command-window will inform you of the progress. After a little while it'll be finished and you can press any key to exit. (Now, where's that 'any'-key again?) Just to be safe, wait another 10 seconds before removing the usb-cable from the phone, cutting its power. Now install the battery again.
Step 4: (Start ClockworkMod, install su.)
Hold volume-down while pressing the power-key. Keep both buttons pressed until you see an LG-Logo with a bluish halo. Now you should be in the regular ClockworkMod. Choose «Install Zip from sdcard» (by navigating to it with the volume-buttons, then select it with the power-button) then «choose zip from sdcard». Navigate to where you put SU303.zip, select it and confirm. After a short while it'll inform you it's done. Press back-button, select «reboot system now.»
Step 5. (Install Busybox.)
Now go the market and install and run «Busybox installer» by JRummy16. Grant it root access when it asks, click OK bottom right, scroll to the bottom of the next screen and click Install. Voila, you're done. You can now do all the things you usually do, like restoring Titanium Backups etc.