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The U.S. neither condones nor prohibits dual citizenship. In a sense, it's ambivalent as to that issue. Instead, people are simply viewed as either Americans or foreigners (hence the term "Alien", which means "foreign"). If you're an American, you're subject to U.S. law, including entering the country with proof of U.S. citizenship and paying taxes on your worldwide income. For foreigners or "aliens" who wish to naturalize and become Americans, the question is whether their own countries allow dual citizenship.
Immigration lawyers encourage U.S. permanent residents to apply for U.S. citizenship as soon as possible, whether after 5 years (regular track), 3 years (living in marital union with an American), or 1 year (served in the U.S. armed forces). However, many countries do not allow dual citizenship, which necessitates a case by case risk/benefit analysis.
As an example, Germans will lose their citizenship upon reciting the U.S. Oath of Allegiance, and becoming Americans, unless they apply in advance for a retention permit, or "Beibehaltungsgenehmigung". (It's true, while most countries don't care that you recite the U.S. oath of allegiance, Germans take it pretty darn seriously when you openly express to a U.S. immigration officer that you are renouncing "all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty", Germany included.)
Most Asian countries do not allow dual citizenship either, and Chinese nationals can be especially wary of losing Chinese citizenship because they're not likely to ever recover it if they later have a change of heart. Did you know that China barely naturalizes anyone in a given year? It's true. They have too many citizens; well, people, to be more specific. Plus, inheriting Chinese property from parents as an American triggers lots of unfavorable taxes in China.
India allows former citizens to reside and work in India with a special OCI card (Overseas Citizenship of India), while Japan and South Korea have only slightly relaxed the prohibition on dual citizenship. (Japanese nationals can retain a second citizenship acquired through birth or marriage.) Meanwhile, the Indonesian government is under pressure to relax their dual citizenship prohibitions also.
Many Asians procure U.S. citizenship but keep it secret and take steps to ensure that their governments don't find out about it. For a listing of countries that support or reject dual nationality as of 2001, see Citizenship Laws of the World (OPM 2001). While this list has not been updated, it may be a good starting point for your research as to what you own country's position was on dual citizenship back in 2001.
Why become a U.S. citizen? See my prior blog post, "Why Bother Applying for U.S. Citizenship?" U.S. permanent residents should speak to a U.S. immigration lawyer about becoming Americans once eligible to do so.