You guys, as a lawyer and someone with a vested interest (I am now a dual citizen, too), I have done some research on this.
Section 215(b) of the INA [8 USC § 1185(b)] requires, in general, that any US citizen who is either leaving or entering the US must be in possession of a valid US passport. This requirement applies even in the case of a dual citizen travelling between the US and his other country of citizenship. A person in such a situation may therefore need to take two passports for the trip -- one from the US, and one from the other country.
Certain exceptions to the US passport requirement are spelled out in Section 53 of Title 22 of the Code of Federal Regulations [22 CFR § 53].
A US passport is not required, for example, when travelling between the US and adjacent countries in the Western Hemisphere, such as Canada or Mexico (but not Cuba). However, other proof of US citizenship (such as a birth certificate or a naturalization certificate) -- together with a photo ID -- may be required at the border in lieu of a US passport. Note that a driver's license or other photo ID, by itself, is not considered acceptable proof of US citizenship by border officials.
Another exception to the US passport requirement exists for young children with dual US/other citizenship. A dual-citizen child under age 12 may travel without a US passport if he or she is listed as a dependent in the foreign passport of an alien parent. In such a situation, other evidence of the child's US citizenship (such as a birth certificate) will have to be shown when returning to the US. Note, however, that this exception may be of limited usefulness if the airline being used chooses not to honor it.
I would also note the following:
When a native-born American citizen takes on a foreign citizenship by a routine oath of naturalisation, the US government applies a "Uniform Standard of Evidence" not to question the intent of such a person to retain US citizenship. This new tolerance of dual citizenship is about 14 years old, and it is not written in a statute. Rather, it is a mere change in State Department policy. Note that the relevant statute on US citizenship continues to state that naturalisation abroad is a "potentially expatriating act"!
In any event: I see as risky any acts by US expats who have become citizens of other countries by naturalisation if the act might be seen as inconsistent legally with intent to retain US citizenship, such as travelling into the US on a non-US passport. Indeed, I think trying to travel into the US on the other passport would mean you would have to fill in a form requesting a visa waiver, thus implying legally that you could be subjected to visa requirements in certain circumstances - it almost sounds like documentary evidence of your intent NOT to retain US citizenship, doesn't it??!!.
I would counsel Americans who are dual citizens, whether by birth in the US, by birth outside the US to American parent(s), or naturalisation, to use only a US passport when travelling to the US. It is not worth the hassle to try to skirt this.