There are some places that will only accept Kuna (for example, to go up on the city walls, it's 90 Kuna for adults, 30 Kuna for youth, no Euro accepted). If you use your CC in stores, they convert Kuna to US Dollars.
The ship exchanged for Kuna onboard and would take Kuna bills back if you exchanged with them. They had increments of 50 Kuna when we exchanged on the inaugral. We had some left over and just gave it to our guide as part of his tip. We paid him in Euro per his agreement but then just added to the tip we were already giving him by giving him our leftover Kuna bills.
We heard a lot in Italy about how switching over to the Euro killed their economy and they are still recovering from that. Our guide was telling us how Croatia has hopefully learned from Italy's experience and will be more cautious in the switch to Euro for currency and that it will be a long while (as Woody has indicated). Everything we read before going was that Euro was accepted as a courtesy and nod to the importance of tourism to Dubrovnik - our guide said that 70% of the employment in Dubrovnik is directly related to tourism.