In Europe, there are a handful of different Y-DNA "clans" as well as mtDNA "clans". You can find out which of these you belong to, and some clans are more common in some countries than others, but pinning down your 'recent' country of origin is easier with a paper trail, augmented by DNA.
For example, my mtDNA which is passed from mothers to their children, points to Ireland, Scotland, and England for my mom's mothers' line, but originally (thousands of years ago) started in what is now northern Italy. Check out some websites for more detailed stuff. Family Tree DNA has a good one, as does Oxford Ancestors.
You may get lucky, if there's a surname project underway for you surname, you may get the test for a discount, or as I did, for free.