Flying with your instrument
If you plan to fly with your instrument you need to be aware of the specific airline's restrictions on checked and carry on baggage. Check through this link or with your travel agent. Most dulcimers are technically considered oversized.

Carry On
If your instrument is small enough, you may be able to carry the instrument on the plane and stow it in the overhead or in the closet. If you attempt this, make sure you are able to enter the plane early as adequate overhead and closet space becomes hard to find the later you board. Also make sure that all tuning wrenches and tools are in checked luggage or they will be confiscated. If you use this strategy though, you should be prepared to gate check the instrument as a fall back position. That means that your instrument can't be in just a soft case but in a case which will offer some protection for the bridge face of the instrument. If you gate check an instrument in just a soft case, you are asking for trouble. If you are traveling on smaller commuter airlines, gate checking will work with minimal risk to the instrument assuming you have a case that provides some bridge protection.

Most airlines now allow only 62" (total length + width + depth) and 50 lbs as the maximum measurement and weight that can be checked in baggage without incurring extra charges. Most of the airlines charge extra ( anywhere from $50-$150) if your instrument and case exceed those limits although from people's experience, enforcement is not consistent from airline to airline and airport to airport so it is possible there may not be any extra charges.

International flights generally have the same size and weight allowances but may cut down on the number of pieces that can be checked. If you know what airline you are going to by using, do check their specific requirements (see above link).

If you are looking for possible cases that would facilitate checked instruments, I recommend you consider my EXPO flight cases.

Strategies
Find a small instrument that will either fit in the overhead, the closet, be gate checked or incur the least oversize charge when checked with baggage. My 3/13/12, 15/14 with 7/8" string spacing, and 3/16/15/8 are all quite small. You might want to check out how one customer deals with flying with a 3/15/16/8. Domestically the alternative is to ship the dulcimer ahead of time via UPS or FedEx where you'll pick it up either at a friends or at a shipping station. You would ship it back to yourself when you return.