The Irish parted ways with manager Giovanni Trapattoni on Wednesday following back-to-back defeats in World Cup qualifiers to Sweden and Austria that look likely to have ended their chances of advancing to Brazil next year.
While some fans have demanded more from the nation, Dunne believes that they cannot expect too much from Ireland.
"If you look at the squad we have, we have to be realistic in what we do," Dunne told reporters. "We very rarely put out a team full of Premier League players and if we do, I don't know how many of them play every week, so it's hard.
"We have a small pool of players to pick from and for us to have been in a playoff and then in a finals.
"All right, we didn't cover ourselves in glory this time, but it has been three campaigns where we can say we have given it a really good shot."
Dunne returned to the starting lineup for their recent qualifiers having been out for a year with injury problems.