At 0400 BST on Thursday about 42,000 mail centre staff and drivers up and down the country began a 24-hour strike.
On Friday, about 78,000 delivery and collection workers are expected to walk out.
The prime minister called on both sides to get back to the table.
"This strike will be self-defeating if all it means is that less people use the Royal Mail," Gordon Brown said.
"This strike is solvable and I believe that management and workforce can resolve it."
Union leaders have blamed Business Secretary Lord Mandelson and Royal Mail chiefs for the failure to reach a deal.
But ministers say the suggestion that an agreement was vetoed is "fantasy".
Mail strike 'self-defeating' says Brown
A BBC poll of more than 800 adults found that twice as many people sympathised with the postal workers than with the Royal Mail management.
Dave Ward, CWU deputy leader, said his message to Royal Mail and Lord Mandelson was to "get your hands dirty" and meet the union at the conciliation service Acas.
He also criticised Lord Mandelson, describing his actions as "vindictive".
The business secretary said that trading insults would not help resolve the dispute.
"Politicising it or dramatising it is absolutely useless," Lord Mandelson said.
"I'm not sure that my intervening would be welcome by both sides. Both sides within the Royal Mail itself need to resolve this."
'Period of calm'
Royal Mail condemned the strikes, calling them "damaging" and said that the launch of extra action was "appalling but sadly not surprising" .
"We had an agreement with the CWU drafted on Tuesday night, around midnight," said Paul Tolhurst, operations director at Royal Mail.
"We were expecting frankly for these strikes to be called off and for a period of calm."
He added that the company had had more than 80 meetings with the CWU.
David Ward of the Communication Workers Union urges management to meet for talks
"There are huge areas where we both agree on what we need to do. There are some areas we still need to work on," he said.
The decision to go ahead with strike action came after several meetings between Royal Mail and the union this week.
Some companies, including John Lewis and Amazon, have said they are using alternative delivery services.
The Federation of Small Businesses estimates that 70% of the 4.8 million small firms in the UK rely on the Royal Mail for their post. It says that every postal strike costs its members, who have up to 20 staff, £300 each.
The FSB's Stephen Alambritas said that his members were struggling to find alternatives to the Royal Mail.
"Our members have tried to shift to other companies - they've found them expensive and actually a number of them have been turned away because the other service providers can't meet the demand," he told the BBC.
Delivery is absolutely key for online retailers, and customers need to be confident that their orders will arrive in time for Christmas
"We really want these postal workers and the Royal Mail to get back to work."
If the strikes continue for a prolonged period, contingency plans for delivery of hospital appointments and medical test results have been drawn up, MPs were told earlier this week.
And it has emerged that the Ministry of Defence may charter extra aircraft to ensure serving troops get their Christmas post.
Postal workers formed picket lines outside 37 mail centres in the UK.
Colin Elcome, on a picket line in Cardiff, said the workers had terms and conditions to protect, as well as an industry and a public service.
"A lot of this is payback time because we defeated them on privatisation. Rather than building the industry up and the public service up, it seems that some of them want to smash it into the ground," he said.
Royal Mail says union 'walked away'
There were about 20 people - including pickets and supporters - outside the Royal Mail's Mount Pleasant sorting office in London.
The mood was peaceful, with no sign of any tension or police presence, correspondents said.
About 20 people had gathered outside a sorting office in Leeds, while police watched about 30 workers staging a protest at Liverpool's main sorting office.
Delivery vans could be seen entering and leaving the Birmingham mail centre, despite the strike and the presence of about 40 workers demonstrating with banners.
The Conservatives said a complete change of culture was needed.
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