You can use Lightning to get a calendar that's fully integrated with Thunderbird, but what about Google Calendar synchronization? Well, there's an extension called "Provider for Google Calendar", which lets Lightning read and write events to Google Calendar.
For older Ubuntu versions (11.04, 10.10 and 10.04), you can install Thunderbird 10 (soon, version 11 will be available) using the stable Thunderbird PPA:
sudo add-apt-repository ppa:mozillateam/thunderbird-stable
sudo apt-get update
sudo apt-get install thunderbird
sudo apt-get install xul-ext-lightning xul-ext-gdata-provider