* Triple-digit heat blankets southern third of the country
* State of emergency in Ohio, Virginia, West Virginia, D.C.
* Restoring power could take up to a week in some areas
WASHINGTON, July 1 (Reuters) - A heat wave baking the eastern United States in record temperatures is set to continue on Sunday after deadly storms killed at least 12 people, downed power lines from Indiana to Maryland and left more than 3 million customers without power.
Emergencies were declared in Washington D.C., Ohio, Virginia and West Virginia on Saturday because of damage from overnight storms, which unleashed hurricane-force winds across a 500-mile (800-km) stretch of the mid-Atlantic region.
President Barack Obama authorized the Federal Emergency Management Agency, or FEMA, to coordinate all disaster relief efforts in storm-ravaged Ohio.
The storms' rampage was followed by roasting temperatures that topped 100 Fahrenheit (38 Celsius) in several southern cities, including Atlanta, where the mercury hit 106 degrees (41 Celsius), according to Accuweather.com.
Heat advisories remained in effect into Sunday across the southeast and lower half of the Mississippi valley, with "triple-digit temperatures expected across the southern third of the country," the National Weather Service said.
"It is very unsafe outdoors for those susceptible to these extreme conditions," it warned in a statement.
Power crews worked into the night to try to restore service to homes and businesses, and officials said in some areas the job could take up to a week. Utilities in Ohio, Virginia and Maryland described damage to their power grids as catastrophic.
Six people were killed in Virginia in storm-related incidents, and more than 1 million customers were left without power in the worst outage not linked to a hurricane in the state's history, said Bob Spieldenner, a spokesman for the Virginia Department of Emergency Management.
Two Maryland residents died in the storm - one struck by a falling tree in Anne Arundel County, the other electrocuted after a tree crashed into a house in Montgomery County - said state emergency management agency spokesman Edward Hopkins.
In New Jersey, two cousins aged 2 and 7 were killed by a falling tree in a state park. And in eastern Tennessee, heat was blamed for the deaths of two brothers, ages 3 and 5, in Bradley County. They had been playing outside in 105-degree Fahrenheit (41-degree Celsius) heat.
In Maryland, about 800,000 customers lost power, with outages reported throughout the state, Hopkins said.
Ohio, where one storm-related death was reported, faced similar difficulties. Outages hit two-thirds of the state with about 1 million homes and businesses left without electricity. Governor John Kasich said it could take a week to fully restore power.
West Virginia was also hard hit by storm-related outages, with about 614,000 customers without power, said Terrance Lively, spokesman for the state emergency management agency.
Further north, the storm caused outages from Indiana, where 135,000 customers lost power, to New Jersey, where Atlantic County declared a state of emergency and at least 206,000 customers were without power.
Blazing temperatures in New York, where thermometer readings were in the 90s (32-38 Celsius) on Saturday, came ahead of a threat of strike by 8,500 Consolidated Edison workers over a new contract.
Early Sunday morning, the company locked out its unionized workers after contract talks broke down, both sides said, raising the possibility of power cuts during the heat wave.
The company asked to extend negotiations for two more weeks, it said, but the union refused. In response, the firm told union members not to report for work on Sunday.
The action increased the risk of power outages if the heat wave puts extra strain on the electrical grid for New York City and suburban Westchester county.
"Both sides are far apart," said company spokesman Mike Clendenon. "We asked the union to extend the talks for two weeks but they refused."
"We can't operate the system reliably for customers if the union can still call a strike at a moment's notice," he said.
John Melia, a spokesman for the Utilities Workers Union of America (UWUA) said that as of 2 a.m. Sunday (EDT) its 8,500 ConEd power workers were locked out.
"ConEd took the extreme measure of locking out its unionized workforce putting the city of New York and Westchester county in peril during a heat wave."
Both sides continued talking for over an hour after the midnight Saturday deadline expired, but failed to reach a settlement over a new contract for the company's unionized workers. A major sticking point in the contract was ConEd's plan to phase out defined pensions.