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 centre.
The internet shutdown happened hours after the military blocked access to Twitter and Instagram to stop people mobilising 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 organisations urged internet providers and mobile networks to challenge the blackout order, Reuters news agency reported.
Human rights group Amnesty International called the shutdown "heinous and reckless" and warned it could put the people of Myanmar at risk of human rights violations.
The military has not commented. It temporarily blocked access to the internet following the coup on 1 February.