Connect second monitor to MBP and new Cinema display
Updated with more detail below...
In order to power your LG LCD your best bet may be to use an external device to power the monitor. Matrox's Dual Head 2 Go would fit your needs. It has been updated for Thunderbolt equipped Macs.
You may run into some strange side effects of using a system like this though. Up to now Matrox's DH2GO causes a Mac to recognize the two monitors connected as a single display. This means your main menu is split across the monitors and dialogs that pop up may show up somewhere between the two.
Conjecture Lies Below
I don't know how Thunderbolt may change this (either at a hardware or software level) but it's likely that it could as Apple will need to pay attention to this type of setup now that they are advertising a MacBook powering two displays via Thunderbolt.
They may also use it as a way to push people toward two Apple Displays instead of a hybrid setup like you suggest.

Update
Apple has added a knowledge base article on this topic which may answer many of your questions. You can find it here: http://support.apple.com/kb/HT4885
How the displays work will depend on the exact computer you have (including the screen size of a MBP) check the details included in that article.
Also, there's an article in Macworld with some details on what happens if you connect an older monitor to Thunderbolt chain. In short for some strange reason you may need another Thunderbolt peripheral to get both your monitors working correctly.