Edit: This is now obsolete, an extension has been written for this: https://extensions.gnome.org/extension/153/nothingtodo/

There's no configuration value for this, it's hard-coded in at the moment, but it's easy enough to change:

gksudo gedit /usr/share/gnome-shell/js/ui/panel.js

Find the line that reads:

this._dateMenu = new DateMenu.DateMenuButton({ showEvents: true });

And simply change true to false

Then restart GNOME Shell by pressing Alt-F2 and entering r

A useful resource for reference (see components section): http://blog.fpmurphy.com/2011/05/more-gnome-shell-customization.html

Edit: For anyone who's interested, here's the end result:

You could try: https://github.com/vintitres/gnome-shell-google-calendar

Few steps are enough:

1. Setup dependencies:

   sudo apt-get install python-gtk2 python-dbus python-gdata python-iso8601 python-gnomekeyring

2. Download it from: https://github.com/vintitres/gnome-shell-google-calendar/zipball/master

3. Extract it in any folder you like.

4. Run gnome-shell-google-calendar.py:

   /path/where/you/extracted/gnome-shell-google-calendar/gnome-shell-google-calendar.py

5. You should beforehand have a online account set up with Google (see: What are online accounts and how do they work?). You should get prompted with this text:

   A list of existing accounts is below. If you do not see a list of accounts, then you first need to add one. For more information, see http://library.gnome.org/users/gnome-help/stable/accounts.html

   0. yourname@gmail.com Please choose the Account:

6. Enter the number of your account and the program will start to syncronize.

7. If you don't want to run it manually everytime, you should add gnome-shell-google-calendar.py to startup applications (see this answer: How do I manage applications on startup in GNOME 3? , insert in the field "command" the path of gnome-shell-google-calendar.py ).

Done!