A near-total internet blackout is in effect with connectivity falling to 16% of ordinary levels, said the monitoring group NetBlocks Internet Observatory.
In the main city, Yangon, crowds chanted "Military dictator, fail, fail; Democracy, win, win".
Police with riot shields have blocked the main roads into the city center.
The internet shutdown happened hours after the military had blocked access to Twitter and Instagram to stop people mobilizing for protests. Facebook had been banned a day earlier.
Many users had evaded the restrictions on social media by using virtual private networks (VPNs) but the more general blackout severely disrupted that.
Civil society organizations urged internet providers and mobile networks to challenge the blackout order, Reuters news agency reports. Human rights group Amnesty International called the shutdown "heinous and reckless".
The military has not commented. It temporarily blocked access to the internet following the coup, on 1st February.
Protester: 'We have to end it now'
On Saturday, protesters including factory workers and students called for the release of those detained by the army, including elected leader Aung San Suu Kyi.
They marched through the streets of Yangon as city buses sounded their horns in support. Bystanders flashed the three-finger Hunger Games salute, which has become a symbol of defiance against authoritarianism in the region.
Demonstrators gave police roses and bottles of drinking water, calling on them to support the people not the new regime.
"We're here to fight for our next generation, to free them from a military dictatorship," one woman at the rally told Agence France-Presse. "We have to end it now."
Myanmar – also known as Burma – has remained mostly calm in the aftermath of the coup, although some demonstrations have been held in different parts of the country.
The military authorities are hunkered down in the capital, Nay Pyi Daw, and have so far avoided direct engagement with the protesters.
The BBC's Nyein Chan in Yangon says the Burmese know very well the violent crackdowns that the military is capable of. The country was ruled by an oppressive military government from 1962 to 2011.
But now that people have had time to digest what is happening, they are finding different ways to get their voices heard, our correspondent says.
Australian academic detained
Ms Suu Kyi is under house arrest, according to her lawyer. Police documents show she is accused of illegally importing and using communications equipment – walkie-talkies – at her home in the capital.
Meanwhile, an Australian academic, who was an economic adviser to Ms Suu Kyi, was detained in Yangon. Sean Turnell told the BBC he had been confined to his hotel and did not know what he might be charged with.
The coup took place as a new session of parliament was set to open, following November's landslide election win by Ms Suu Kyi's National League for Democracy (NLD) party.
Many Burmese watched the events unfold in real-time on Facebook, which is the country's primary source of information and news. But three days later, internet providers were ordered to block the platform for stability reasons.
Following the ban, thousands of users were active on Twitter and Instagram using hashtags to express their opposition to the takeover. By 22:00 local time (15:30 GMT) on Friday, access to those platforms had also been denied.
There was no official word from the coup leaders but AFP reported it had seen an unverified ministry document that said the two social media sites were being used to "cause misunderstanding among the public".
A spokeswoman for Twitter said the ban undermined "the public conversation and the rights of people to make their voices heard". Facebook, which owns Instagram, called on the authorities to "restore connectivity".
Myanmar at a glance
Myanmar is a country of 54 million people in South East Asia which shares borders with Bangladesh, India, China, Thailand, and Laos.
It was ruled by an oppressive military government from 1962 to 2011, either directly or indirectly, leading to international condemnation and sanctions.
Aung San Suu Kyi spent years campaigning for democratic reforms. A gradual liberalization began in 2010, though the military still retained considerable influence.
A government led by Ms Suu Kyi came to power after free elections in 2015. But a deadly military crackdown two years later on Rohingya Muslims sent hundreds of thousands fleeing to Bangladesh.
It triggered a rift between Ms Suu Kyi and her previous supporters in the international community after she refused to condemn the crackdown or describe it as ethnic cleansing.
But she has remained hugely popular at home, shown in her party's landslide win in the November election.