An Post has advised customers not to post any items with immediate effect due to industrial action.
Staff who maintain key equipment in the four An Post mail centres are taking strike action over cuts in pay.
The 36 staff in question are employed by contractor IO Systems, but are retained by An Post to maintain the automated postal processing machinery at the centres in Dublin, Cork, Portlaoise and Athlone.
In a statement, An Post said it "greatly regrets this inconvenience to our customers and urges the parties involved to resolve their differences through the normal industrial relations procedures".
It added that post office retail services are operating as normal.
It is understood that all automated post processing machinery at the Dublin centre broke down overnight, which had an impact on mail sorting.
In Athlone, the machinery for sorting large objects also broke down.
The machines in the two remaining mail centres in Cork and Portlaoise were functioning normally.
An Post said this afternoon it has not issued protective notice to staff but noted that if there is no mail coming into the system, there is no work for its employees.
Communications Workers' Union official Seán McDonagh said An Post managers refused to repair the machines as that was the work of the IO Systems employees.
He also pointed out that the object of the new rosters sought by IO Systems is a cost saving of €100,000, but the disruption to An Post services resulting from the dispute will cost much more.
IO Systems has said it deplores the strike, and accused the CWU of total contempt for its obligations.
In a statement, the company said the action was in total breach of the CWU/IO Systems agreement in respect of the CWU obligation to refer for binding decision to the Labour Court "all and any unresolved disputes".
It said IO Systems staff, instructed by CWU, are now in default of their conditions of employment to work their newly-assigned shift requirements as determined by the Labour Court.
The Department of Social Protection said it has contingency plans in place for delivering cheques to customers, should the dispute impact on next week's mail delivery.