Can I trace family history for free?

I'm trying to trace the genealogy in my family back to Germany. The name I have to go on is Thieme - which is a German surname - if that can help anyone help me. I wanted to know if there is a place I could look it up and trace it back for free.
Update: My great-great-grandparents are the only ones who immigrated from Germany to America. To be more specific, I want to find the Thiemes that lived in Germany and staid their up until post-WWII.
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