Google has recalled travelling staff members to the US after an executive order from President Donald Trump restricting entry for nationals of seven Muslim-majority countries.
Syrian refugees are banned from entry until further notice, the order says.
Nationals of six further countries, including Iran and Iraq, will be banned from entering the US for 90 days
The entire US refugee admissions programme is suspended for 120 days, and a lower cap on numbers introduced.
On Saturday several Iraqi passengers and a Yemeni national were prevented from boarding a flight at Cairo airport bound for New York, despite holding valid visas for the US.
Google has told the BBC it is concerned about the order and any measures which could block great talent from the US.
The new restrictions will have a major impact on technology companies that hire skilled staff from all over the world on special H1-B visas.
There have been reports that "green card" holders, who are legal permanent residents of the US, being prevented from getting on flights. However, green cards are not specifically mentioned in the executive order, and so the status of green card holders remains unclear.
CAIR advised non-US citizens, including permanent residents, from the seven countries to plan to delay all international travel for at least 90 days.
Mr Trump said the measures detailed in his executive order would "keep radical Islamic terrorists out of the US".
But rights groups say there is no link between Syrian refugees in the US and terrorism.
Mr Trump signed the order on Friday, which was International Holocaust Remembrance Day.
The president's statement to mark that occasion, on the 72nd anniversary of the liberation of Auschwitz, made no mention of Jews or anti-Semitism.
Threat to technology: Joe Lynam, BBC business correspondent
The sudden and severe immigration restrictions imposed on passport holders from seven Muslim countries could seriously demonstrate the law of unintended consequences. The president wants to restrict some Muslims but the effect could be to damage America's most important and profitable sector: technology.
Google has recalled around 100 of its affected staff from overseas. Microsoft has warned its shareholders that curbs on immigration could have a material impact on its business.
The technology sector relies heavily on highly skilled and well-paid workers from overseas on H1-B visas. If there's a risk, however small, that that brainpower could be restricted in some way or scare off others who may feel unwelcome, the big tech companies may have to rethink where they place their key staff in future.
The American-Arab Anti-Discrimination Committee also advised nationals from the affected countries but living in the US not to travel abroad.
A petition organised by academics in the US opposed to the executive order includes the signatures of 11 Nobel laureates.
Some Republicans have welcomed Mr Trump's announcement, including the Speaker of the House of Representatives, Paul Ryan, who said it was "time to re-evaluate and strengthen the visa vetting process".
The new policy is also affecting visits from relatives.
An Iraqi journalist living in the US, Mohammed al-Rawi, posted on Facebook, saying his father had been turned away from a Los Angeles-bound flight in Qatar.
Jamal Abdi from the National Iranian American Council told the investigative journalism organisation Pro Publica: "We are inundated with calls and questions of how this is going to affect people."
Facebook founder Mark Zuckerberg posted a lengthy note, saying he was "concerned" about the president's executive orders, and noting that he, like many Americans, is the descendant of immigrants.
Under Mr Trump's wide-ranging executive order, all refugee admissions have been suspended for four months.
The text of the order was released several hours after it was signed. Other measures include:
- A ban on refugees from Syria until "significant changes" are made
- A 90-day suspension on anyone arriving from Iraq, Syria, Iran, Libya, Somalia, Sudan, and Yemen, except certain visa categories such as diplomats
- Priority for future refugee applications from those persecuted for their religion - but only if the person is part of a minority religion in their home country
- A cap of 50,000 refugees in 2017 - less than half of the upper limit under Mr Trump's predecessor, Barack Obama
The order also said all immigration programmes should include questions to "evaluate the applicant's likelihood of becoming a positively contributing member of society."
Other measures include a broad review of the information required from all countries to approve a visa; a review of visa schemes between nations to ensure they are "truly reciprocal" for US citizens; and the immediate suspension of the Visa Interview Waiver Programme.
But the document says exceptions could be made on a case-by-case basis.
During the election campaign, Mr Trump suggested a "total and complete shutdown of Muslims entering the United States until our country's representatives can figure out what is going on".
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