The little boys, aged nine, were found dead in their home just outside the town of Charleville in the north of the county shortly before 5pm on Thursday.
They have been named locally as Thomas and Paddy O'Driscoll. It is understood the twins were stabbed.
About an hour later, their older brother, believed to be aged in his early 20s, was found dead in a forested area near a river just outside the town of Buttevant, about 10 miles from the scene of the first deaths.
Garda detectives are not seeking anyone in connection with his death.
A local priest spent about two hours with family members during two visits in the hours after the deaths were discovered.
Fr Tom Naughton, of the Holy Cross Parish in Charleville, said: "We prayed together. We comforted them and assured them, especially of the community here in the area, that we were with them.
"It's clear that anybody who suffers a tragedy is going to be upset and hurt at this time but it is quite raw."
It is understood Garda detectives are investigating whether the three died in an apparent murder-suicide.
The alarm was raised when a member of the family arrived home and discovered the youngsters in the detached, pink and white bungalow just off the main Cork-Limerick road, an area of Charleville known as Deerpark.
A major police search was then launched for a third member of the family.
His body was discovered at a wooded area near the river Awbeg at Castlelands less than a mile from Buttevant.
Gardaí said the two young boys were pronounced dead at the house.
Two police checkpoints were erected on the northern and southern sides of the main road about two miles from the house, believed to be the family home of the two boys.
A number of emergency services vehicles, including ambulances, remained at the scene several hours after the discovery, along with forensic specialists and uniformed officers manning roadblocks.
Post mortems on all three sons were carried out in the city's university hospital.
It is understood the O'Driscolls were settled members of the travelling community and had adopted children and had been living in the area for many years.
Health chiefs said specially trained counsellors were working in the town to support families and children affected by the tragedy.
Community health workers with the Travellers of North Cork Association were also in the town supporting members of the community, relatives and friends.
Counselling and support services are being set up in the area by the group alongside the HSE to help the immediate and extended family and they are expected to remain in place for several days into next week.
The Bishop of Cloyne William Crean extended his sympathy to the family and urged people to pray for them.
"I speak for our faith community, and for all people of good will, when I say how shocked and greatly saddened I was to hear of the tragedy that has befallen the O'Driscoll family," the bishop said.
"Such loss of human life is extremely difficult to comprehend. I pray that the Lord will strengthen the family during this terrible time.
"I ask the priests and people of the diocese to pray for Thomas and Helen, for their surviving children, their extended family, and for the community of Charleville."
Distraught parents Thomas senior and Helen were overcome with emotion as black hearses carrying their nine-year-old twins Tom and Paddy pulled out from the side of the house on to the main road to Cork.
In tears, the couple pressed against the windows of the cars as they edged away.
As counsellors spoke to school children, family, friends and members of the local travelling community, the principal of the school Tom and Paddy attended paid tribute.
Sheila Cagney, of Banogue National School just over the county border in Croom, Limerick, described the twins as "loving, energetic and full of fun".
"This is a terrible tragedy for the family, the school and the whole community. We are deeply saddened by this event," she said.
"Our sympathy and thoughts are with the family and friends at this terrible time."
Ms Cagney said the focus of the staff in the days and weeks ahead was on the pupils in the school who are trying to come to terms with the loss of young friends.
"We will help the children to the best of our ability," she said.
Gardaí are appealing for witnesses or anyone with information to contact them or any other Garda station or to use a dedicated confidential line in the Republic on 1800 666 111.
If the murder-suicide suspicions are confirmed, it will be the second incident of its kind in Ireland in just over six weeks.
Two brothers died in a tragedy at their home in rural Sligo in late July - nine-year-old Brandon Skeffington was found with stab wounds in the family home at Banada, Tourlestrane near Tubbercurry, before the body of his older brother Shane junior, 21, was found in a shed beside the property.
It was the first murder-suicide to occur in Ireland for more than a year.
A study released last August examined 19 similar incidents over a 13-year period from January 2001 to the end of June last year.
The report revealed that on average one murder-suicide incident happens per year and 46 people died in the period under review.