Here is what I have found since I first reporting my problem. It turns out this is a Catalyst Driver issue. On a different computer running Ubuntu 10.04 and with a Radeon HD 4890 I was able to get solid graphics performance with no screen tearing using the latest open source ati driver. However, on the same machine with 10.04 and the Catalyst driver, I am seeing screen tearing.
Just to recap my other machine has a Radeon HD 5870 and I am seeing tearing with the Catalyst driver. Unfortunately, however, the open source ati driver does not currently support the Evergreen series of GPUs, so I am stuck with this pitiful driver. I am now hopeful that my problems will be solved later this year when 5xxx series cards are supported in the open source graphics stack.
I'm looking forward to Ubuntu 10.10 and beyond. I think this problem won't be fixed until that time and any future AMD/ATI GPUs I would advise staying away from until the open source driver can support them out of the box. It is my understanding that this screen tearing issue is rooted in something fundamental to the proprietary Catalyst driver and is not going away any time soon.