It is fairly easy to see the country of origin of Euro coins, but what about notes? The letter in front of the serial number tells you which country the note is from. And here is the list:
D: Estonia
E: Slovakia
F: Malta
G: Cyprus
H: Slovenia
L: Finland
M: Portugal
N: Austria
P: Holland
S: Italy
T: Ireland
U: France
V: Spain
X: Germany
Y: Greece
Z: Belgium
Luxembourg notes, however, are a bit different (imagine having unanimity in the EU, outrageous idea). According to the ECB:
Uncirculated euro banknotes issued by the Banque Centrale du Luxembourg bear the code of the central banks of the countries where the banknotes for Luxembourg are produced