ATMs could be the cheapest, but not if your bank (like many US based banks) charges hefty fees for foreign currency withdrawls and/or use of some other bank's ATM. Also, €1000 is likely to be larger than the default ATM withdrawl limit for many ATM / debit cards (so you may need multiple cards attached to different accounts if you go this route).

If you know someone who recently came back from a € using country, ask if s/he has any excess to sell to you.