After a few hours of trading in Tokyo, the important Nikkei 225 index and the broader Topix index had both fallen by around 3 percent each.
South Korea's Kospi index initially lost 2.3 percent but then recovered somewhat and was down 1.6 percent at lunchtime.
The gloomy tariff atmosphere was also felt in Hong Kong, where the Hang Seng index retreated 1.7 percent.
In Shanghai and Shenzhen – which, like Hong Kong, are affected by Trump's China tariffs reaching at least 54 percent on many goods – the cloud was somewhat lighter as the composite indices fell by 0.2 and 0.7 percent, respectively. Beijing has previously promised market-friendly stimulus to counter Trump's tariff policy.
Japan faces Trump's announcement of 24 percent American import tariffs, South Korea 25 percent.
