First, open the "Buzz" label in GMail. Click on your list of followers and block all of them. Click on "connected sites" and remove any connections.

Now edit your Google profile (if you have one) and remove any information that you don't want public (including "display my full name"): http://www.google.com/profiles/me/editprofile?edit=b

Finally, go to the footer of the GMail site and click "turn off Buzz."