I think that there are a great many things that Dumbledore had at his disposal that your average Azkaban inmate wouldn't have enjoyed. (Others have made some of these points already but I think they also missed some of Dumbledore's options out so look at this as my version of a complete list).
- An ability to do wandless magic.
In order to escape Azkaban, the first obstacle you have to overcome is your lack of a wand. Most wizards perform some rudimentary, random wandless magic as children but then become overwhelmingly dependent on their wands. Exceptionally powerful wizards, however, are able to channel their magical abilities so that they can perform specific spells without a wand, even as children. This was true of Voldemort.
"His powers, as you heard, were surprisingly well-developed for such a young wizard and - most interestingly and ominously of all - he had already discovered that he had some measure of control over them, and begun to use them consciously. And, as you saw, they were not the random experiments typical of young wizards: he was already using magic against other people, to frighten, to punish, to control."
(Half-Blood Prince, Chapter 13, The Secret Riddle).
Dumbledore is at least Voldemort's equal in terms of power and ability. If an untrained Riddle could do the things he could without a wand then just think what an adult Dumbledore at the pique of his powers could do. Charming metal bars, destroying walls, conjuring a boat - I have no doubt that he could do these things without a wand. Most inmates aren't that talented and so Azkaban probably wouldn't have spells to counter the use of wandless magic from the inside.
- A conviction of his own innocence.
The second obstacle you have to overcome is the constant sapping of hope and joy by the Dementors. The only known prisoner to have escaped from Azkaban by himself was Sirius Black, who was able to resist the power of the Dementors. As vap78 says, Black survived because he knew that he was innocent of the crimes of which he'd been convicted.
"I think the only reason I never lost my mind is that I knew I was innocent. That wasn't a happy thought, so the Dementors couldn't suck it out of me...but it kept me sane and knowing who I am...helped me keep my powers."
(Prisoner of Azkaban, Chapter 19, The Servant of Lord Voldemort).
If we look again at the charges that Dumbledore was arrested on, we can see that he was totally innocent.
"Weasley!" cried Fudge, now positively quivering with delight, "Weasley, have you written it all down, everything he's said, his confession, have you got it?...The bit about how he's been trying to build up an army against the Ministry, how he's been working to destabilise me?"
(Order of the Phoenix, Chapter 27, The Centaur and the Sneak).
The two charges are building up an army against the Ministry and working to destabilise Fudge. Dumbledore has done neither of those things so would have benefited from the same righteous bubble of protection against the Dementors that Sirius had. Thus he would have kept his identity, his powers and his sense of hope - all of which he would have lost if he'd been guilty and all of which he'd need to pull off an escape.
- A crazily-loyal phoenix that can burst into flames.
As RicoRicochet points out, Fawkes would have come in pretty handy in an escape. Indeed, using Fawkes was how Dumbledore escaped from his office. Fawkes can Apparate inside Hogwarts, which shows that he isn't to be waylaid by the magical constraints that apply to humans. If Azkaban had charms and defences to prevent Apparition by wizards they probably wouldn't keep Fawkes out. Once Fawkes finds Dumbledore, he can simply Disapparate with him to a safe location. Needless to say, phoenixes are exceedingly rare and most prisoners don't have the option of calling upon one to escape on.
- A working knowledge of the magic of house-elves.
As Oak says, Dumbledore understands the ways of house-elves in a way that most wizards don't. Maybe Azkaban has protections against incursions by elves. Maybe they don't. If they don't then it wouldn't be that hard for Dumbledore to call upon an elf that was loyal to him (e.g. Dobby) and Disapparate with him to a safe location. Most wizards (not just Voldemort) overlook the magic of house-elves and so wouldn't consider that they have a magic that they themselves do not.
- Insider connections in the Ministry of Magic and the criminal underworld.
OK, some of the inmates probably have connections in the criminal underworld. But not only can Dumbledore call in favours from a whole network of undesirables like Mundungus Fletcher but he also has access to friends in high places in the Ministry of Magic. Someone high up in the Department of Magical Law Enforcement (say an Auror like Nymphadora Tonks or Kingsley Shacklebolt) could probably ask to visit Dumbledore in Azkaban. They could then give Dumbledore a wand and try to break out together. This is the high-risk option that also involves revealing the confidante's affiliation to the Order.
If all else fails, having a huge, semi-civilised giant on your side's not going to hurt, right?