Whenever you travel abroad, you've gotta use common sense. In general, most of Southeast Asia is significantly safer than the US, but you should still be cautious when in big cities or areas of instability.
Keep your valuables safe. When I travel I always carry a money belt for our travelers cheques and airline tickets as well as a hidden neck pouch for our passports and any cash or cheques we might need for the day. Never leave valuables in your room. It's not a bad idea to leave your purse or wallet at home - pickpockets and thieves could grab them up easily.
Don't stick out like a sore thumb. The more you look like a clueless target, the more likely you'll become a clueless target. If you're entering areas not commonly visited by foreign tourists, don't wear shorts or tank tops unless you want to take a chance offending the locals. Fanny pouches are a bad idea - they make you an obvious tourist. If you need to look at a map, it's not a good idea to stand in the middle of the street unfolding a big paper map - way too vulnerable. Keep a small map in a book or a local newspaper so when you open it you won't look too out of place.
Carry your stuff wisely. If you've got a camera with a strap or a daypack, don't carry them swinging in one hand. Someone could easily rip them away from you. Keep them over the shoulder or around your neck. Never leave them strapped around a chair when you're sitting down somewhere. You could either forget them or have them stolen while you're munching on some pad thai.
When using a taxi, know your fare upfront. Many taxi drivers like to take advantage of unsuspecting tourists. The meter rate is usually the cheapest way to go, so before you get in, either insist on using the meter or negotiate a price. Sometimes the driver will quote you a fair price; sometimes you'll get scammed. Try to find out what a reasonable fare should cost. Many travel books like Lonely Planet will mention such fares in individual chapters. In general, three-wheeled bicycle taxis (cyclos) are the cheapest form of transportation, followed by shared pickup trucks (songthaews) and three-wheeled taxis (tuk-tuks in Thailand, jumbos in Laos). Full size car taxis are always the most expensive.