The GEORGE Surname project is coordinating tests on the Y chromosome, which is passed from father to son unchanged, except for occasional mutations. To track your GEORGE ancestors, you would need to find male cousins with the GEORGE surname. I believe the tests are an exciting way to identify family lines descended from a common ancestor and distinguish families which share a surname, but no common ancestry.
There is also a test available on mitochondrial DNA, which is passed from mother to daughter. However, the GEORGE surname project is not sponsoring mitochondrial DNA testing.
The GEORGE Surname web site referred to in the original message has links to Family Tree DNA and other sites. Family Tree DNA provides more details on DNA testing and has links to other informative sites on DNA testing.
I do not believe that Family Tree DNA is associated with the Mormon Church. If curious, you could send an inquery to Family Tree DNA.