Under the Vienna Convention of Road Traffic, drivers licenses from one country are formally accepted by other countries - as long as both countries are signatories to the convention. The US is NOT a signatory to the convention, although most (all?) European countries are. However, most US states accept most all foreign licenses, with the exception of the state of Georgia, which requires an international driving permit.
However there then becomes the problem of proving you have a valid license. If your home-country license is in English, or contains an English translation on it (as many EU licenses do) then that is all that you will need.
If your home-country license is not in English then you will need to make use of one of the other features of the Vienna Convention - the "International Drivers Permit". This is not a license in itself, but in effect translation of your home-country license into English (and several other languages). The IDP must be obtained in your home country before you travel - exactly how will depend on your country but Google should be able to tell you how to obtain one. Note that the IDP is NOT a license in it's own right - it's just a translation of your license, so it is only valid when used in conjunction with your real license.
The above is true both for the legal requirements for driving, as well as for rental car companies who will need to see your English drivers license, or your non-english drivers license plus the IDP when you are picking up the car.