There are no problems with cash exchange in Odessa like anywhere else in Ukraine - exchange booths are scattered all around, many of them work till late night. US dollars, German marks and Russian rubles are exchanged by majority of exchange booths. But some of them operate with many different stable currencies, up to 25 currencies. Please, note damaged banknotes might be not accepted.

Situation with credit cards and traveler's checks has grown much better lately - many banks in Ukraine started operating with them. Though only credit cards that banks operate with are Visa and MasterCard. American Express, Discovery, Diners are not accepted on the territory of Ukraine. 

For cashing credit cards and traveler's checks you will need your travel passport. 

Also, there are several ATMs in Odessa. 

Here is the list of banks in Odessa that operate with traveler's checks and credit cards.