Chanting and waving flags, they have blocked roads in the capital Khartoum and around the country following the takeover.
On Monday coup leader Gen Abdel Fattah Burhan dissolved civilian rule, arrested political leaders and called a state of emergency.
Soldiers opened fire on crowds and reportedly killed 10 people.
Gen Burhan is expected to address the media on Tuesday afternoon. Earlier, he sought to justify the takeover by blaming political infighting.
The coup has drawn global condemnation. Diplomats told AFP news agency the UN Security Council is due to meet on Tuesday to discuss the crisis.
Troops are reported to have been going house to house in Khartoum arresting local protest organisers.
The city’s airport is closed and international flights are suspended. The internet and most phone lines are also down.
Central Bank staff have reportedly gone on strike, and across the country doctors are said to be refusing to work in military run hospitals except in emergencies.
Civilian leaders and their military counterparts have been at odds since long-time ruler Omar al-Bashir was overthrown in 2019.
US Secretary of State Antony Blinken said the military’s actions “are a betrayal of Sudan’s peaceful revolution”. The US has halted $700m (£508m) in aid.
After a night of protests, demonstrators remained on the streets on Tuesday morning, demanding the return of civilian rule.
“Civilian rule is the people’s choice,” they chanted as they set up barricades of burning tyres. Many women are also taking part, shouting “no to military rule”.
The protests continue despite troops opening fire on demonstrators on Monday.
One wounded protestor told reporters he was shot in the leg by the army outside the military headquarters, while another man described the military firing first stun grenades, then live ammunition.
Pictures from a hospital in the city showed people with bloodied clothing and various injuries.
World leaders have reacted with alarm to news of the military takeover.
The US has joined the UK, EU, UN and African Union, of which Sudan is a member, in demanding the release of political leaders who are now under house arrest in unknown locations.
Among them are Prime Minister Abdalla Hamdok and his wife, along with members of his cabinet and other civilian leaders.
BBC Arabic’s Mohamed Osman reported from the capital that a special security unit of the military went to the prime minister’s home early on Monday morning, and tried to persuade Mr Hamdok to agree to the coup, but he refused.
Gen Burhan, who was head of the power-sharing council, said Sudan was still committed to the transition to civilian rule, with elections planned for July 2023.