Ivo Flipse blogged about staying fit while traveling so I'll enumerate some of the points from his post.
Jump ropes are small, portable, and don't require much space. Leg exercises such as lunges and squats will work out the biggest muscles in your body and will more calories than working a different part of your body.
If you have a smartphone or iPad, you can use the Workout of the Day app which sends you a different workout each day. The workouts require no gym and focus on using your own body weight.
Jogging is another activity you can, but it is largely dependent on where you will be and whether you want to carry running clothes and shoes. You run the risk of getting lost or ending up in a less than desirable location unless you plan your route carefully. Outside of running, walking in itself is perfectly fine. If you want to track your walking distance, there are pedometers you can use to do so. If your locale features tall structures, climbing to top also provides a workout in addition to some spectacular views.
Burpees are excellent to perform in small spaces as well. The steps to do them are:
- Begin in a standing position.
- Drop into a squat position with your hands on the ground.
- Kick your feet back while lowering yourself with a pushup.
- Return your feet to the squat position while straightening your arms.
- Leap up as high as possible from the squat position with your arms overhead (you may clap your hands above your head at the peak of your jump) Image from Look Like an Athlete.com
Also see the following questions for more workout ideas, which Ivo referenced in his blog post.