Dubrovnik is a nice town in the far south of Croatia. The old town is a UNESCO world heritage site and nearly over run by tourists in the summer months, it is extremly difficult to CS in this place.
Hitchhiking north out of Dubrovnik - does require some patience, all the rest areas seem to be located on the coast, making it difficult to find a good area. The spot I found worked, although I waited an hour. Go the the main bus station, in the north. Cross the road and keep walking north and look for steps leading up the hill side. If find yourself under the bridge, you have gone too far. If you follow the path, it will lead you all the way up to the E65. Then, cross the bridge and continue across to the other side, where you will find a rather small pull in spot. This is the only spot I could find for hitchhiking north out of Dubrovnik. Make sure you stay behind the barrier and make yourself as visible as possible, because the pull in spot is not long and the drivers can't see it from far away. Halfway up the path to the E65 that you would have taken, is an awesome stealth camping spot. Ok, so when you get about halfway up the path (probably a good idea to check it out during the day, was a little scary when I found it at 1.30am, drunk) if you look to your left, you should be able to see a metal railing, at the bottom of a fairly steep slope like 3 metres long or something. Be careful. Then turn to your right, with the railing on your left and walk along. There are some pretty annoying bushes that will probably scratch you, but hey, it seems to keep other people out. And then you should be finding yourself with about 1 to 2 metres between the railing and the cliff behind, just enough space to lie down, in a secluded spot with a great view. Enjoy ;-) However, be very careful and use a torch, as there are sudden drops in many places and walking here without light is dangerous.
Option 2: Nothing complicated. There are bus stops on the Jadranska Cesta. I stood at the one opposite to the Lodpudska Ulica with a HUGE sign "Ploče". Literally 1 minute later, the first car passed by pulled over and took me to Ploče.
East towards Montenegro 8
Hitchhiking out of Dubrovnik - which is the last major town in the south of Croatia - does not seem that difficult, although it might require quite some luck. Just climb up to the coastline road and hitchhike on the E65. A couple of kilometers after the city, there is a scenic view with a cafe and a large parking space, large enough to stop a truck. IMPORTANT! If you are already on the part of E65 close to the old town, DONT move any further east! For the next 2,5 km there is not much space to walk - not to mention to stop for the cars because of crash barriers, and you will have to walk it all. Its a very scenic - but also not very safe walk, in case you want to give it a try.
Most drivers, though, are heading mainly towards either airport or the small town of Gruda. It is still about 10 km from there to the Montenegro-Croatian border crossing between Herceg-Novi (Montenegro) near Gruda (Croatia). Since the traffic is pretty low on that road - especially in the late afternoon/evening - you will need some luck to find someone crossing the border and willing to take you. Most probably you will be picked up by some tourists.
Along the coastline road there are plenty of places where you can put up a sleeping bag, though few places large enough for a tent. However, be very careful and use a torch, as there are sudden drops in many places and walking here without light is dangerous.
Sleeping in a Hostel in Dubrovnik is an option which costs around EUR 13 per night. You can be unlucky in the summer months where it might be booked out if you appear without reservation. Check pages like hostelworld.com or hostelbookers.com.
It seems to be very hard to get someone who will host you. Because Dubrovnik is loved by tourists, couchsurfers from this town are spammed by requests and will probably not answer.
Port of Dubrovnik
500 metres from the main bus station in the direction of the old town is the "port of Dubrovnik", which is open 24 hours a day. In summer, the place may be a bit noisy and inconvient to get some sleep, however, in other times of the year it would generally provide a good night's sleep especially if it's cold or raining.