This is actually what I recommend to anyone who wants to visit Dubrovnik, to make a detour and visit the Bay of Kotor in Montenegro. It's heaven on Earth and it's only an hour's drive south of Dubrovnik. The whole ride around the bay is very picturesque and I spent much of the time with my jaw dropped at this majestic landscape.
The bay is known as Europe's southernmost fjord, but it's in fact a submerged river canyon. The dramatic backdrop of mountains and the crystal clear, and calm, waters of the bay, the cultural heritage, made UNESCO list the Bay of Kotor as one of their protected World Heritage sites.
Also, to get a sense of how much this place impresses tourists, here is what the romantic poet, Lord Byron said of Montenegro, 'At the birth of our planet, the most beautiful encounter between the land and the sea must have happened at the coast of Montenegro. When the pearls of nature were sown, handfuls of them were cast on this soil.'