Some COs and supervisors (like the Prefectural ALTs) ask JETs to submit an itinerary when they travel abroad which includes flight information, where you will be staying, contact information, etc. This is a good idea because then if something happens, someone has information on where you are supposed to be and has a starting place to begin the process of trying to check in with you.
If your CO does not ask for a travel itinerary (and perhaps even if they do), it is a good idea to let someone, a fellow JET or good friend, know what your plans are.
Have copies of important documents
It is a good idea to make a copy of your passport, visa, residence card, etc. and store the extra copies in a suitcase separate from your original copies. It will make the process of getting back to Japan much smoother if you do end up losing track of these things.
KEEP RECEIPTS OF ALL HOSPITAL EXPENSES
If anything happens (though hopefully not!), you will likely end up having to pay upfront for all your hospital costs, etc. Keep those receipts. You should be able to make a claim after you return to Japan and recompense some of those costs, but it is only possible with proof.
EXTRA TRAVEL INSURANCE
Unfortunately, your things will not be covered when you go traveling. It is not a bad idea to look into extra travel insurance just in case.