Investors were also disappointed by falling commodity prices and a mixed finish on Wall Street on Friday despite positive employment news. There was also some frustration regarding mainland Chinese shares, which opened to losses after a weeklong holiday.
"That worried investors that we might have hit the top of the recent rally and we might see some correction in the short term," said Jackson Wong, vice president of Tanrich Securities in Hong Kong.
Hong Kong's Hang Seng fell 0.6 percent to 20,879.40. South Korea's Kospi lost 0.8 percent to 1,980.33 and Australia's S&P/ASX200 dropped 0.3 percent to 4,480. Mainland China's Shanghai Composite Index shed 0.8 percent to 2,070.52 and the smaller Shenzhen Composite Index lost 0.8 percent to 847.39.
Benchmarks in Singapore, Taiwan and Thailand also fell. New Zealand's rose. Markets in Japan were closed for a public holiday.
The World Bank cut this year's growth outlook for developing Asia-Pacific economies to 7.2 percent from its May forecast of 7.6 percent. The bank cut its forecast for China, the region's biggest economy, to 7.7 percent from May's 8.2 percent. The bank cited weak global demand due to the lackluster U.S. recovery and Europe's recession.
Andrew Sullivan, principal sales trader at Piper Jaffray in Hong Kong, said protests in Spain and IMF concerns over Greece continue to worry investors.
Tens of thousands of people marched in 56 Spanish cities Sunday to protest government budget cuts in a country experiencing its second recession in three years and record high unemployment.
The government has pushed through nine straight months of tough austerity measures that have prompted Spain's 17 regional governments to slash spending in health care and education.
Officials from the European Commission, International Monetary Fund and European Central Bank are currently in Greece assessing the country's progress in fulfilling the terms for receiving aid.
If their report doesn't clear the way for the payment of the next 31 billion euros ($40 billion) tranche of the country's bailout, Greece could be forced to default on its debts and perhaps leave the euro. Greece has warned that it will run out of money next month if it does not receive its next scheduled loan.
Greece is caught in a deep recession, and has unemployment of nearly 25 percent. On Friday, the country's statistical authority said the economic contraction in 2010 and 2011 was even worse than earlier thought.
German Chancellor Angela Merkel is to visit Greece this week for the first time since the European debt crisis erupted. Merkel is unpopular in Greece because her government has been instrumental in pushing Athens to make austerity cuts in exchange for its bailout loans.
Falling commodities prices hurt resource and metals-related shares. Hong Kong-listed Jiangxi Copper Co. fell 0.9 percent. Zijin Mining Group, China's biggest gold miner, tumbled 3.1 percent. Australia's Newcrest Mining Ltd. lost 3.3 percent.
On Friday, the U.S. Labor Department said the unemployment rate had declined to 7.8 percent, its first dip below 8 percent in nearly four years. The decline from 8.1 percent the month before was bigger than economists had expected.
Stocks rose on that news, but the gains didn't last. The Dow Jones industrial average edged up 0.3 percent to close at 13,610.15. The Standard & Poor's 500 index fell marginally to 1,460.93, and the Nasdaq dropped 0.4 percent to 3,136.19.
Benchmark oil for November delivery was down 41 cents to $89.47 per barrel in electronic trading on the New York Mercantile Exchange. The contract closed down $1.83 to $89.88 per barrel on the Nymex on Friday.
In currencies, the euro fell to $1.2989 from $1.3025 late Friday in New York. The yen rose to 78.53 yen from 78.69 yen.