Since the start of March there have been 47 such incidents.
Four of these happened in Cork - three in West Cork and one in North Cork in the last number of months.
In one incident in Leitrim, a dog attacked seven sheep on a farm.
The owner was identified and had to pay damages to the farmer and the dog then had to be put down.
In another incident in Meath, a farmer shot two dogs as the dogs had killed six sheep and injured two more.
Garda Inspector James White said dog owners can be held liable for damages and face prosecution if their pet attacks animals on another person's land.
"Farmers are also within their rights to shoot any animal that is worrying their livestock," he added.
Insp White urged dog owners to keep their pets under control.
"Whilst you might think your dog is friendly and placid at home, they can cause horrific injuries to other animals.
"We would ask that everyone ensures their dog is kept under control as nobody wants to see animals being harmed," he said.
Between March 1 and June 29, incidents also occurred in Sligo, Galway, Tipperary, Roscommon, Longford, Kildare, Donegal, Wexford, Cavan, Monaghan, Westmeath, Mayo, Limerick, Kerry, Kilkenny, Carlow, Laois and Offaly.
Prosecution under Section 9 of Control of Dogs Act 1986 could result in one month in prison and/or €600 fine.