Try to look as if you know where you are going, this may not be so easy if after all it is the first ever time in that country, but even still look confident as IF you know.
Carrying maps around looking perplexed, and stopping to look at monuments or buildings, obviously shows that you are from out of town, and may attract the wrong type of attention.
Using travellers cheques, will always be helpful, as will using a credit card for any purchases made so that you can take advantage of the additional insurance offered.
Do NOT pull large amounts of cash out of your pocket. This will catch attention, no matter what country you are in. It sounds so silly, but it is amazing how many people pull out a mound of cash to be some small item.
Try to think about where you are going that day and carry sufficient cash for that and any unforeseen extras, plus a card. That should cover all you need and will limit any losses if anything did happen.
Do NOT accept drinks from anybody that you have just met, especially if in dubious surroundings or do not know, they could be laced with any type of concoction.
Try not to go on "wonderful trips" or to a "super shopping centre" rides with people who approach you in the street. This could lead anywhere, and will probably cost you a lot more than by taking a normal taxi. Ask your concierge or read through this site.
Check on your first day whether you need to reconfirm your next flight, if so do it then. Some airlines do not require this anymore but it is still worthwhile calling them so that they at least have your contact details, in case the flight is delayed or whatever.
Do not carry your passport around, leave it in the safe in your hotel. A photocopy will suffice if local law states that you need to.