For your paternal grandmother's mother, find a cousin who is a female grandchild of hers maternally to compare with them. It would have to be in the direct female line coming down from her.
Y DNA goes from father to son only. In other words you get it from your father, his father, his father etc
Mitochondrial DNA goes from mother to both sons and daughters but only the daughters pass it on to their children. In other words you get it from your mother, her mother, her mother etc.
I wanted to get my paternal grandmother's Mitochondrial DNA and I had a cousin whose mother was her daughter to take the test. Since she was my paternal grandmother I don't have her Mitochondrial even though my father did.
You have 23 chromosomes from each parent and Y & Mitochondrial come from the one each that determines your sex. They are used in genealogy as they both go back in a straight line virtually unchanged.
The other 22 chromosomes each are made of your Autosomal DNA which you get 50-50 from both parents but it is too complicated to use to determine most relationships. It is used for paternity testing of a female.
This company is the oldest and has the largest database for Y & Mitochondrial testing. If you get a "we're sorry" message just click on "Home." If you want to email them and ask question just click on "feedback."
http://www.familytreedna.com/Default.aspx?c=1
This website gives a good explanation of DNA
http://www.smgf.org/pages/how_it_works.jspx
Edit: Another thing you can do is to prove ancestry from your great grandfather. A direct male line descendant from a brother of your grandmother and compare it with a direct male line descendant they have from their great grandfather.
Back many years ago adoptions weren't recorded like they are today. A woman might have had a husband desert her or leave her widowed and when she married any children she had by her previous marriage might have taken the name of her new husband.