An extreme left-wing group is suspected of carrying out the campaign and police are looking for five men in their 20s.
The packages were sent to embassies in Athens, international organisations and European leaders including Germany's Angela Merkel.
Greek police said the airmail service was being suspended to allow checks.
The country's Prime Minister, George Papandreou, said he would be "merciless to those who attempt in vain to rock social peace with terrorist acts and hurt our country's image internationally during a very difficult period".
A private courier plane, found to be carrying a suspect package from Athens and addressed to Italian Prime Minister Silvio Berlusconi, was rerouted to Bologna airport in Italy on Tuesday night.
When examined, the package caught fire. No-one was hurt, Italian officials said, and the contents were taken for analysis.
On Tuesday evening, two parcel bombs were destroyed in controlled explosions at Athens airport's cargo terminal. They were addressed to international police organisation Europol in the Netherlands and the European Court of Justice in Luxembourg.
On Wednesday morning, reports said Athens police were investigating a suspect package at the Argentine embassy.
Parcel bombs exploded on Tuesday at the Swiss and Russian embassies in Athens, and several suspicious packages were destroyed.
A suspected bomb was destroyed at the Bulgarian embassy and another, posted to the Chilean embassy, in a van.
No-one was hurt in the blasts, which came a day after four parcel bombs were found in the city.
The BBC's Malcolm Brabant in Athens says it appears to be a co-ordinated attack by an extreme left-wing group.
Suspect packages either exploded or were set off in controlled explosions at the Swiss, Russian, Bulgarian embassies in Athens on Tuesday. No-one was seriously injured.
Another suspected device addressed to the Chilean embassy was found in a delivery van outside the Greek parliament and destroyed.
German officials said a parcel bomb sent to Chancellor Angela Merkel's office had come from Greece.
"It seems this is a continuation of yesterday's attacks and that Greek guerrillas are behind it, but we are still investigating," police spokesman Thanassis Kokkalakis told Reuters news agency on Tuesday.
On Monday, police arrested two suspects and found two bombs, one addressed to French President Nicolas Sarkozy and the other to the Belgian embassy.
Two other bombs - one addressed to the Mexican embassy in Athens and one addressed to a Dutch embassy - were also discovered.
Terrorism experts suspect the co-ordinated campaign is the work of a group called the Conspiracy of the Cells of Fire, which is trying to spark revolution in Greece during the current period of austerity.
The parcel bombs have raised anxiety levels in Greece in the run-up to this weekend's vital local elections, our correspondent says.
The elections are seen as a referendum on the socialist government's handling of the economic crisis, and Mr Papandreou has warned he may call a general election if his party is soundly defeated.