The authorities said that about 42,000 residents had been driven from their homes by the flood. Most of the damage was in Aceh's Tamiang district, on the northern tip of Sumatra island.
Floods killed 17 people, with almost 50,000 people made homeless, in neighbouring North Sumatra province, officials said.
Landslides triggered by the rains killed another 21 in the province's Muarasipongi district, Hashim Nasution, deputy mayor of Mandailing Natal, North Sumatra, said on Elshinta radio.
Nasution said local residents had just returned to their homes after fleeing last week's earthquake. "We are still trying to locate more people. We received a report that at least four people are still missing in the area," he said.
Mustafa Abu Bakar, Aceh's acting governor, said that army helicopters and navy ships had been deployed to deliver aid and help in the rescue efforts in the remote areas.
He said about 5,000 people were still trapped in the Pinding district, with access roads cut off. Communications to the area were also down.
The authorities have blamed heavy rains as well as the effects of deforestation for the destruction. Lack of adequate forest cover leaves the ground less able to absorb excess water.
Sulaiman said some of the waters in Aceh had begun to recede, leaving behind thick mud that complicated rescue and aid efforts. The Bener Meriah region was cut off and the authorities were waiting for a helicopter to survey the damage, he said.
He said the government and aid organisations had sufficient supplies of food, tents and medicine but transport capable of reaching remote areas remained a problem.
Almost exactly two years ago, on December 26, 2004, Aceh was hit by the Indian Ocean tsunami, which left about 170,000 people dead or missing in the province.