My favorite place in Croatia is Mala Hiza, but it is not in Zagreb (2 hours away, but worth the trip for an amazing rustic meal).
In the city, I can recommend Trilogija (daily menu) and Vinodol (more traditional)
Interesting insights and many half-facts. First of all, as a historian and a Turkish resident I must say that the notion of the Persian roots is very wrong. While there is some influence of Persian culture to the cultures of Anatolia, Persian rulers only ruled for around 300 years, during the Achaemenid Empire. Turks as such only came to Anatolia in the 10th century, both Seljuk and later Ottoman, and they brought with them eating traditions that are a little different than the ones the indigenous cultures had.
Modern Turkish cuisine is mainly influenced by its Ottoman heritage, and depending on the region in some places local traditions are really strong.
Turks don't consume seafood that much, besides fish and mussels, and occasional fried calamari. Shrimp dishes are a rare treat for those who can afford it, but to my utter shock most people here find seafood disgusting, especially octopus, crabs etc.
Turks use a lot more butter and dairy in general. Greek food is lighter, heavier on typical Mediterranean spices, basil, oregano, rosemary. Turks prefer sumac, isot, aci biber and other kinds of pepper, and they love the sheep tail fat.
Lebanese cuisine on the other hand feels fresher and lighter than Turkish, aromas are more clearly pronounced.
Personally, I can't believe that Turks don't have their own version of falafel, since many Turkish dishes have an equivalent in Levantine cuisine.
I feel blessed to live in this part of the world where I have an easy access to all three cuisines. I'd personally choose Greek for seafood and meat, Lebanese for vegetarian dishes and salads, Turkish for desserts and mezze, as well as soft cheeses and olives.
I know it is too late now, but maybe it will help someone else.
In Croatia it is extremely important to know where to go, cause unfortunately it is one of those countries where homecooking often exceeds affordable dining options in restaurants. (high-end restaurants serve great food, but are very expensive, pretty much overpriced).
In Bosnia, definitely try cevapi (Zeljo and Ferhatovic Petica are usually a safe choice). Bosnia is very cheap, and food is very similar to what you would find in Turkey but with less vegetables and more starches. A lot of pite (pies) and turkish style deserts (in sugary syrup).
Zagreb has been undergoing a gastronomic revolution and there are a lot of new restaurants that set new standards for all true foodies in the region.
In Split, Nostromo at the fish market is a very nice place. There are so many new restaurants it is impossible to recommend just one. There are also plenty of traditionak konobas to check out in the city centre.
In Vis, the best places are Pojoda and Vatrica in Vis, an informal restaurant on a farm in Zena Glava, Bako in Komiza (dont miss out the tour to the Blue Grotto).
In Dubrovnik if you get bored of the grilled fish and dalmatian cuisine, the food is superb in Oyster and Sushi bar Bota and the veggie place Nishta ("nothing" :)) )
Also, don't miss visiting Istra - the cuisine there is spectacular. And many of the best Croatian restaurants are there (Kukuriku, Le Mandrac..)
This a great website to check out: http://www.tasteofcroatia.org/eateries/