Are there any US laws that a US citizen MUST obey while traveling, even if those laws do not exist in the country he is traveling to?

(This is going to over-simplify things a little, but...)

Technically, as a US citizen you are covered by the laws of the US regardless of where in the world you are. However with very few exceptions, when you are outside of the US you are outside of the jurisdiction of those laws. ie, if you're a 19 year old US citizen and you're drinking in Australia then you're not breaking any law - simply because there is no law against 19 year old US citizens drinking whilst in Australia.

There are, however, exceptions.

The US government can - and has - passed laws which include a jurisdiction outside of the US. The most well known of these is the "PROTECT Act of 2003", which makes it illegal for a US citizen to partake in illicit sexual conduct abroad with someone under the age of 18. In effect, the intent of these laws was to stamp out "sex tourism", and many other countries have passed similar laws.

Generally, if you break a US law even though you're not in the US - you can be prosecuted if the victim is a US person, even if what you did was legal at the place where you did it.

As mentioned by @Doc, there are explicit laws regarding sex tourism, taxes, volunteering for military service, and other stuff.

Some things may be legal, but not wise. American teenagers are notorious for getting drunk everywhere they go just because they can. Now imagine yourself a bunch of kids coming into your neighborhood just to get drunk, and that's what the locals would think of you for doing that. Its not illegal, but it is still damn stupid.

My understanding is there exists a presumption against the extraterritorial applicability of United States law unless there is explicit language in the law to indicate it was the intent of Congress for the law to have applicability outside the territory (and designated special areas) of the United States. This is not to say the United States does not have the ability to enfore its laws outside its territory; the United States can and in some cases does, normally violations of US criminal codes, enforce its laws on actions occuring in other sovereign States.