As a would-be American citizen, I experienced such suspicions and cautions first-hand. Upon my immigration to the US in the early 1970s I found it entirely reasonable and prudent for my adopted country to check me out and demand certain conditions in exchange for the highly sought-after and much-appreciated US citizenship. I gladly complied with the conditions of citizenship: a working knowledge of the English language, a basic understanding of civics, the promise not to become a burden to the state, and above all, loyalty to my chosen homeland. This was and still is accomplished with the oath of allegiance. The most important aspect of qualifying for naturalization as an American citizen, the oath requires the new citizen to renounce any foreign allegiances and to support and defend the constitution. This oath is quite clear and unambiguous and taken voluntarily by a new citizen.
These citizenship requirements were a small price to pay for the freedom I enjoyed, the ability to shape my own destiny, the unlimited potential I could pursue with perseverance and drive. Although lengthy, I never resented the thorough background investigations to which I was subjected before being granted a commission in the US Air Force and eventually giving me access to Top Secret intelligence information. To me it was the embodiment of limitless opportunities offered by my adopted country - a country I considered my own many years before ever setting foot on American soil.
At the core of American values is freedom of choice - we chose to live here because we identify with the American way of life. We also have the right to leave anytime we no longer feel comfortable here, unlike many other countries that lack of this option in. Not coincidentally, many of our immigrants are from just those countries. But the oath of allegiance should guide all of us -- native-born and naturalized citizens!