Can I drive using my U.S. (New York State) license in the European Union (if the answer depends on the country, please answer for Czech Republic)?
You could simply get an International Driving Permit. This, coupled with your US driver's license, is recognised in over 70 countries.
The American Automobile Association and the American Automobile Touring Alliance's National Automobile Club are the only entities in the United States to issue IDPs.
As for recognising your country license, each EU country is different. No EU country is obliged to recognise an international license, even if already recognised by a fellow country.
So since you need an IDP for the Czech Republic, I'd recommend just getting it and then you're covered for any European country.
Here's information about driver's license Latvia: http://www-en.csdd.lv/?lngID=EN&pageID=1181212466 Basically it means you can use your US license for up to one year. After that you neet to get Latvian driver's license by passing driving test.