I've been having the same problem and I think I may have found a solution. Note that I've only tested this on one computer and so far this seems to be working.
First you need to install the US English layout and remove it again to make sure it's not set on your user account.
Go to Start->Run and type in
intl.cpl
First go to the "Keyboards and Languages" tab and click on "Change keyboards". This opens a new dialog where you should see a list of installed layouts but US English is most likely not listed.
You need to add the US keyboard by clicking on "Add...", find English (United States), open that and the underlying "Keyboard" by clicking on the + and there select the option "US". Click on "OK", this closes the window and you're back in the "Text Services and Input Languages" window.
Click on "Apply" to install the keyboard layout.
Then select the US English keyboard, click "Remove" and then "OK".
Now go to the "Administrative" tab and click on the "Copy settings" button. Now you should see locale settings for your current account, Welcome screen and new user accounts. It's very likely that the "Input language" value for "New user accounts" is set to "US English"
Under "Copy your current settings to:" select both "Welcome screen and system accounts" and "New user accounts". The values for both "Welcome screen" and "New user accounts" should change to your current settings.
Click on "OK" to confirm and again on "OK" to close the "Region and language" settings.
If this works for others and anyone feels like making a little more graphical version of this, feel free!
Best Answer
Σιμπλζ αδδ τηε Γρεεκ λαζοθτ το ζοθρ Κεζβοαρδ λαζοθτ πρεφερενψεσ. Τηεν αδδ τηε Κεζβοαρδ Ινδιψατορ αππλετ το ονε οφ ζοθρ παλελσ ανδ θσε τηατ το σςιτη λαζοθτσ.
Simply add the Greek layout to your keyboard layouts. Then add the "Keyboard Indicator" applet to one of your panels and use that to switch between layouts. It seems to work for me.
Edit:
Changing the layout via a key combination seems to be a bit flaky:
Choose "Layout options" in the Keyboard preferences dialog and find the group of options labled "Keys to change layout". You will then simply have to try the different options and see which one works for you, both on a system level and a usability level. I guess for the option that says "both alt keys" you will need two Alt keys (I have only one here). On the other hand, I definitely have to Ctrl keys, but the "Both Ctrl keys" option doesn't work for me either.
I understand that you want to avoid cluttering your panel, but my guess is that you will need some sort of indicator of the current layout or you'll find yourself constantly typing in the wrong one. Of course, your mileage may vary.