Giovanni Trapattoni's side will have to overcome Group C opponents Croatia, Italy and the reigning European champions Spain in order to make the last eight, but Dunne is adamant that the current Ireland side can compete with anyone.
"Yes. There's no point in us being here otherwise," Dunne told reporters when asked if he thought the Irish could make it out of Group C.
"We have to have the belief that we can go all the way and win the tournament" - Richard Dunne
"We have to have the belief that we can go all the way and win the tournament - but then every team that's there will believe the same thing.
"We know we are as organised as anyone and as strong as anyone, so it's just about hopefully having that little bit of luck that can help us progress."
The Irish have been training in Tuscany ahead of the tournament and Dunne insists the mood in the camp is positive, despite the news during the week of Kevin Foley's omission from the squad.
He added: "The atmosphere is really good, really positive. We are a close bunch of friends anyway, so the longer we spend together, the better the atmosphere is and we know that when we go into the games, we are going to fight for each other."
There had been concerns about the 32-year-old's fitness after a lengthy lay-off with a collarbone injury during the Premier League season.
But Dunne played down concerns, saying: "I have been training for two months nearly, and I played the last three games, and was then involved on Saturday and again the other night, so I feel fine."