The Montenegrin city of Kotor has, without a doubt, already established itself as Dubrovnik-light. It has all the qualities of i's cousin to the north, but just on a smaller scale, and with about half the tourists, although I'd expect that to change fast. As Dubrovnik gets overcrowded, those sick of the crowds have pushed farther south into Montenegro and have reached Kotor. Luckily, most seem to only come for the day, or at the most, stay for the night.
Kotor has everything you need to be a top tourist destination. It has a stunning location on the Boka Kotorska (for my money a more impressive location that Dubrovnik), an immaculate walled city, sunshine, and a towering fort looming over head. Luckily, the place is so small, that it will never become the destination Dubrovnik is, but for those who make the trip, it can be equally rewarding.
The trick, I discovered by accident, is to arrive nice and early in the morning. As I said before, most tourists come to Kotor for the day from either Dubrovnik or Budva, therefore they don't arrive until lunchtime and leave before sunset. Thus, for the dedicated traveler who arrives in the morning (9-10 am), you can have the city almost to yourself.
Kotor's old city, tucked into a corner of the Boka Kotorska, hidden behind a set of triangular walls has all the charms of Dubrovnik, smooth marble streets, towering cathedrals, funnily named squares (e.g. "Square of Milk"), and plenty of outdoor cafes. What it doesn't have though are hordes of sunburned British tourists prancing around in shorts and piercing your ears with glottal stops. The town is surrounded by 20m-high walls and has only three gates, one complete with drawbridge. Once inside the maze of alleys and corridors are enough to keep you wandering for an afternoon and the churches are awe-worthy, particularly the Cathedral of St. Tryphone. St. Tryphone, easily Kotor's best monument, dates all the way back to 1166, and is a magnificent piece of work.
Wandering around the rest of the Old City, you will find a Maritime Museum, which would be skipable if it weren't for a collection of surprisingly nice paintings, as well as a couple more fine churches.
Although, I highly recommend saving all that for later in the afternoon. If you arrive in the morning, the first thing to do, without a doubt, is head up to the Fortress of St. Ivan. At 1,350 steps, the hike isn't exactly an easy one, and will be much better in the morning when the sun is still hiding behind the mountains. The hike would be a bit more painful if you weren't stopping every 10 feet to take a picture. The views only get better as you go up. About halfway up the steps you'll reach the aptly-named Chapel of Our Lady of Salvation, which was oddly enough built by Black Plague survivors. Once you reach the top, the view will be unforgettable. So sit on the ruined walls, have a drink, and snap away. While the fortress itself is nothing spectacular, I particularly enjoyed sitting there, taking it all in, and thinking about all the people over in Dubrovnik fighting the crowds who won't find a view half as nice as this.
If you do the fortress first, by the time you get down the crowds should be arriving, making it a perfect time for a lunch at one of the hundreds of outdoor cafes. By the time you finish, the crowds should be dissipating and the rest of the city will be yours.
My one regret with Kotor is that I didn't stay overnight. I can imagine that the city at night is serene and peaceful. The city has very few hotels, but there are some inside the old city for surprisingly reasonable rates. To this day I still wish I could have had one night of wandering around the streets of the old city in solid moonlight. It must look perfect. So that is my suggestion to you, stay overnight. Not only will you get to see Kotor at all stages of the day, but you could also get some time to explore nearby attractions on the Boka.