Japan's Nikkei 225 index fell 0.5%, Hong Kong's Hang Seng shed 1% and Australia's ASX 200 was down 0.3%.
On Wednesday, a weaker-than-expected Spanish bond sale saw concerns over the eurozone debt crisis re-emerge.
In the US, markets fell 1% after the Federal Reserve signalled that it may not provide more economic stimulus.
"The major support for the economy and for the financial markets over the past two years has been stimulus, and without it, it's still a question whether these economies can make it on their own," said Bruce Bittles at Robert W. Baird & Co.
The eurozone has been battling with a sovereign debt crisis, which has raised concerns over economic growth and undermined investors' confidence.
There have been fears over the ability of some eurozone economies to repay their debts.
On Wednesday, those fears were fanned further after the Spanish government's latest bond sale fell short of expectations.
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The euro crisis always comes in waves and this one is clearly a new wave”
Holger Schmieding Berenberg Bank
It had hoped to sell up to 3.5bn euros ($4.6bn; £2.9bn) of medium-term bonds, but only managed to sell bonds worth 2.6bn euros.
Spain needs to sell bonds successfully on a rolling basis to meet its existing debt repayments.
To make matters worse, the yields on Spanish bonds already in circulation rose. The yield on 10-year Spanish bonds rose by 0.25 percentage points to 5.7%, the highest level since January.
A higher yield suggests that investors are becoming more worried about Spain's ability to meet its repayments.
Analysts said the disappointing Spanish bond auction was the latest in a long line of concerns for investors and could continue to hurt markets in the coming days.
"The euro crisis always comes in waves and this one is clearly a new wave," said Holger Schmieding, chief economist at Berenberg Bank.
"There is a lot of volatility and today it doesn't feel like it is the bottom already."