The raiders drove into the Metro Bank branch at Babbage Gate, Oakgrove in Milton Keynes at 04:20 BST, police said.
The cash machine was put in a dark 4x4, and the thieves drove off in that vehicle and a white VW Golf.
Police are appealing for anyone with mobile phone footage of the raid to get in touch.
Ray Cardy, who walked past the bank shortly after the raid, said: "The dumper truck had been driven through the front doors, turned round and smashed the ATM from behind.
"The remains of it were on the footpath - the dumper's engine was still running."
A Metro Bank spokeswoman declined to reveal how much money had been taken.

Miss Beckford, who came to the UK from Jamaica aged 17, was also a prominent race equality campaigner.
Marvin Rees said he was "very sad" to hear the news and hers was "a life very well lived".
The former midwife, who was the first black recipient of an MBE in the South West, died at a care home in Bristol earlier this week.
She helped set up the St Pauls Carnival in 1967, was appointed the first Race Relations Officer in Bristol and also worked alongside the bus boycott campaigner Paul Stephenson, to improve racial equality in the city.
Councillor Asher Craig announced Miss Beckford's death and told Bristol24/7: "She was a beautiful woman and had a heart of gold.
"She was extremely stylish and was an amazing networker who had all the skills and expertise of a socialite, which she capitalised on in her role as the entertainments officer on the board of St Paul's Carnival.
"The legacy of Carmen Beckford's contribution towards helping to create a more equal and integrated city must never be forgotten and I will ensure that it lives on."
Miss Beckford is one of "seven saints of St Pauls" who will be commemorated in giant murals, due to be painted by artist Michele Curtis this summer.
The portraits of prominent black Bristolians will be painted on houses or prominent buildings along the route of the St Pauls carnival procession.

Amy Simpson, 18, died when her car left the road near the village of Dalavich, on the banks of Loch Awe, on Tuesday.
Miss Simpson, from Cowdenbeath in Fife, was with four other women passengers who escaped without serious injury.
Her mother Angela Simpson said her "gorgeous girl Amy" was "my best friend, my soulmate and my rock".
She said: "She made me so proud.  One day I will make her as proud of me as I was of her.
"I am absolutely heartbroken and it's hard to find the words at this distressing time to express just how much I will miss her.
"Rest in peace Toots."
Miss Simpson's black Peugeot left the road at about 18:40 on Tuesday and the car ended up in the River Avich.

The 27-year-old's career, which started at French club Montpellier, has taken him to Ukraine and Germany and he spent last season on loan at Ligue 1 side Nice.
"I've joined the biggest club in Turkey and after two years without the title I hope my coming will change that," Belhanda told the club's official website.
He becomes Galatasaray's third signing of the season after Brazilian defender Maicon and French forward Bafetimbi Gomis.
The 2012 BBC African Footballer of the Year nominee made his debut for Montpellier in 2009 and went on to make 144 appearances, scoring 29 goals for the club.
A member of the Montpellier side that won the French Ligue 1 title in 2012, he joined Dynamo Kiev from Montpellier on a five-year deal in July 2013.
He scored 11 goals in 88 appearances for the capital side, winning the Ukrainian League and Cup titles twice.
Belhanda spent the last two seasons on loan with German side Schalke and French outfit Nice.
A former French youth international, Belhanda chose to represent the country of his parents Morocco in 2010 and has two goals in 30 appearances for the Atlas Lions.
Four balaclava-clad men smashed a dumper truck into the front of a bank and escaped with its cash machine.
[NEXT_CONCEPT]
The mayor of Bristol has led tributes to Carmen Beckford, the founder of St Pauls Carnival, who has died aged 87.
[NEXT_CONCEPT]
The mother of a teenager who died when the car she was driving plunged into a river in Argyll has described her as the "perfect daughter".
[NEXT_CONCEPT]
Moroccan international Younes Belhanda has joined Turkish giants Galatasaray from Ukranian side Dynamo Kiev on a four-year deal.