President Assad has promised a series of reforms as his administration faces a wave of anti-government protests.
He sacked the cabinet on Tuesday in an apparent bid to appease public anger after 60 people died in clashes with security forces. Meanwhile, thousands attended the funerals of 15 people killed on Friday in the Damascus suburb of Douma.
The BBC's Lina Sinjab in the capital says the marches went off peacefully, with only a small security presence detected.
The mourners chanted anti-government slogans, calling for President Assad to be toppled, our correspondent says.
Eight bodies were buried on Sunday in Douma; the rest of the victims came from different towns, she adds.
Officials have denied that protesters were shot by troops. They say "armed groups" opened fire from rooftops, not just on the protesters but also on security forces, who they say were among the casualties.
It is difficult to confirm the reports, because of government restrictions on the media.
The wave of protests is the most serious challenge so far to the Assad family's 40-year rule.
The protests were touched off by the arrest of teenagers who wrote anti-government graffiti in the town of Deraa in the south.
President Assad has blamed a foreign "plot" for the unrest, but he has agreed to examine the unpopular emergency law - in place since 1963.
New cabinet Mr Assad issued a decree on Sunday designating Adel Safar to form the new government, said a report from the official news agency, Sana.
Earlier, he issued a decree accepting the resignation of the government led by former Prime Minister Mohammad Naji Otri, it added.
Adel Safar, 58, is a member of the ruling Baath Party's central committee who was heavily criticised for his handling of Syria's recent drought crisis.
He is expected to unveil his new cabinet in two or three days.
In recent days there were also pro-democracy demonstrations in Deraa, a southern city where the unrest began, and Latakia in the north-west.
On Saturday, Syrian security forces arrested dozens of people - mostly in Deraa and Douma.
A joint statement by eight human rights groups, quoted by AFP news agency, said 46 people had been arrested.
Protesters released from custody say that all those detained during the protests were beaten, humiliated and branded traitors by the security forces.
Activists and human rights groups say that between 60 and 130 people have died in clashes in the past two weeks. Officials have said the death toll is closer to 30.