Finding the 'right' answer for you, involves knowing exactly what the question might be. And what you mention really is kind of vague.
If your ancestor of that name immigrated, your goal is to FIND the location where they were born. Your statement that Thieme is a German name is not backed up by what I see in familysearch.org. If you run a quick search on that site, it will clearly show that the name Thieme occurs in numerous places, not just Germany.
Trying to trace SURNAMES is a huge waste of effort. You need to trace the actual persons, by looking in the documents of where they came from.
As for persons of that name who remained in Germany.. again, if you are hoping to find someone who "has the family history".. that may or may not happen.
So.. my advice to you is to come back to your immigrant ancestor, the individual person with that name. Identify YOUR DOCUMENTATION relating to that specific person. I am not saying that he/she did not come from Germany. You may have a document that confirms this. If so, that is good, and that is what you build on.
On the other hand.. if you are trying to build on the idea that "this is a German surname", you are working from the wrong angle. I do see many persons with this name on the immigration lists that originate in the German area.
What is going to be useful TO YOU is to be specific about your own ancestor.