They were the only children of expats Jane and Martin Weekes.
The grandfather of the triplets, Ron Turner, told Newstalk ZB he and his wife would be travelling to Doha tonight to be with their daughter and son-in-law.
They had been contacted by Ms Weekes just after the babies had been found and were told they had died of smoke inhalation.
"She was still just as numb I suppose, and just as empty as my wife and I feel right now and the rest of our family and friends who have rallied around to give us support.
"You go from happy and full life and family to a situation that I can't see how I will ever be happy again,'' he said.
New Zealand journalist Tarek Bazley, who lives 1km from the mall in Doha, was in another part of the mall with his children when the fire broke out.
"The first thing I heard of it was a very benign fire alarm, it sounded more like a doorbell to be honest, repeating in the background,'' he told Radio New Zealand.
After 15 minutes he asked an attendant whether he should be worried, but was told to "sit tight'' and that it was usually a false alarm.
"About 10 minutes later, someone else, a member of the public, raced through this area and said, `Everybody out, you've got to get out now. The other half of the mall is on fire.' That is what he said to me.''
The daycare area was a casual drop-in centre, Bazley said.
"I believe the problem was that once the fire took in the area near the nursery, there was no way to actually escape.''
Both exits were blocked by smoke, he said.
The alarm and sprinkler system appeared to have malfunctioned.
"It's an extraordinary mall in the sense that as you walk down it it has this ceiling that is like a mock sky, so I think in order to achieve that effect, they've actually not put in sprinklers in this area.''
Firefighters had to break through the roof to get to trapped children after a staircase to the first-floor nursery collapsed, an official said.
Dense smoke inside the mall, combined with the fierce temperature from the flames, made reaching the trapped children very difficult, a representative of the civil defence told a news conference.
Happy life in Qatar
The triplets had been going to the daycare three mornings a week for the past couple of months, Mr Turner said.
"That gave them an opportunity to meet more children and be involved in a different environment than home and mum and dad and nannies, and they were so proud to show off their school bags.''
The triplets' bodies would be brought back to New Zealand, he said.
Just last week Ms Weekes wrote on babycenter.com of potty training the trio.
"We have about 7 potties around our living space (it's a really big open plan area and is tiled so we have no issues with accidents ) and in the bathroom to be able to sit one on the toilet with a kiddy seat and 1 or 2 on the pottys if it turned into rush.''
But Ms Weekes said she was not yet confident enough to let the three go without nappies, particularly at night, which she said was miles off her radar.
Ms Weekes also wrote of her love for coordinating the three children's clothes, despite them not all being the same gender.
"Like jeans and different coloured polo or tee shirts, or different styles (Only 1 in a dress!) but same colour. Before they were born I too said I wouldn't but yup it's too cute to resist!''
The three had also discovered Barney, which Ms Weekes described as the "great purple god''.
Villaggio Mall is a popular shopping centre and is the newest in Doha.
It opened in 2006 and has 360,000 sqm of floor space including many well-known Western brands, such as Foot Locker, Pizza Hut, Starbucks and British department store Marks & Spencer, according to the mall's website.
Prime Minister John Key this morning said New Zealand Foreign Affairs staff were travelling to Doha from Saudi Arabia to provide support for the Weekes' family.
"Their family are obviously dealing with the terrible grief that they have at this time. Obviously it's a very tragic day for that family.''
Labour leader David Shearer also sent his condolences to the family.
"I join with all other New Zealanders in expressing my heartfelt sympathies to the parents, extended family, and friends, of the three young children caught up in such a tragic event. The pain of their loss is unimaginable.''
