Shares in Australia were among the biggest losers regionally, with S&P/ASX 200 dropping 3.05% to close at 5,960.60.
In Southeast Asia, Singapore's Straits Times Index dropped 3.56% in afternoon trade.
Overall, the MSCI Asia ex-Japan index fell 1.7%.
Investor reaction to the Fed's recent economic forecast was watched on Thursday. The U.S. central bank kept interest rates unchanged on Wednesday and indicated it does not expect to raise them through 2022. The Fed also expects the U.S. economy to contract by 6.5% in 2020 before expanding by 5% in 2021.
"We think the Fed is gonna keep their foot on the gas until the jobs come back, and there's a lot of jobs to come back," Gareth Nicholson, head of fixed income at Bank of Singapore, told CNBC's "Street Signs" on Thursday. Using the global financial crisis as an example, Nicholson said it took five years before the level of unemployment went back to pre-crisis levels.
"There is a dispersion between the economy and the markets," Nicholson said. "The Fed, I think, is looking more at the economy, they're looking at the unemployment rate which is at record levels. They're looking at a lot of the production and manufacturing which is still, you know, maybe it's turning slowly but it's still struggling."
The U.S. dollar index, which tracks the greenback against a basket of its peers, was at 96.213 after touching an earlier low of 95.95.
"The (Federal Open Market Committee's) decisions does not change our view for the USD," economists at Commonwealth Bank of Australia wrote in a note. "Once market participants digest the FOMC's views, we expect the USD to continue along its depreciating path, consistent with a recovery in the world economy."
The Japanese yen traded at 106.94 per dollar, having strengthened from levels above 109 seen earlier in the trading week. The Australian dollar changed hands at $0.6944 after seeing an earlier high of $0.7004.
— CNBC's Fred Imbert contributed to this report.
Correction: This story was updated to accurately reflect the Thursday moves of China's Shenzhen component.