One of the easiest ways to stay fit and relieve stress is to incorporate a daily walk into your routine. When you're traveling, you don't have to put that daily walk aside.
Walking in a strange, new city can be intimidating and honestly, it can also be unsafe. It's easy to lose your bearings and your way, so unless you know you have a fair amount of daylight in front of you or a specific path that you know you can follow, I don't recommend walking the streets of strange locations alone. However, if you have a basic grasp of the surroundings, a map, or a common path to walk (such as a town's green belt that's common for walking) you can always make the trek out. Like always, make sure someone will miss you should something go awry. If you have a friend that you need to catch up with, grab your cell phone, throw on your blue tooth headset and go. Chatting and walking is usually enough to ward off anyone who might be attracted to you as a target.
If you're not confident with walking the streets, you can always hit the hotel fitness center's treadmill. But one of my other favorite things to do to stay active in a hotel is to hit the stairwells. Again, be safe about it. Still take your cell phone even if you're staying local. Stairs are some of the best cardio exercise around. You can start in the lobby and go to the top floor, then criss-cross using the hallways from stairwell on one end of the hotel to the other side (they usuallly have 2 stairwells on either side) which adds some aerobic walking in between the resistance training that running stairs can provide. If the stairwells aren't appealing, then you can always just walk the hall. People may stare - but who cares. You're probably not going to see them again anyway!
And you can always throw some hand weights or ankle weights into your carryon, too! They can add to your quick workout and make it even more effective.
Exercising while you're on the road can relieve stress and help you sleep. So throw your walking shoes into your carry on and make that walk a priority, no matter where you are.