Many of the issues I have with the systems have already been mentioned, but I'll throw in my two coppers worth:
-First Edition's problem was, again, that the characters were too similar. There's not much to distinguish Flint Fireforge from Caramon Majere, for example, besides their ability scores and weapon types. The level limits for demihumans were too arbitrary and low.
-Second Edition was neutered for Pat Pulling and the other mothers who couldn't be bothered to be responsible parents. No demons or devils, the bad guys always suffer humiliating defeats, etc. General institutional rot creeped in around this time too, with endless series of splatbooks strangling the market. Greyhawk was destroyed, violated by the Greyhawk Wars and From The Ashes.
-Third Edition, as frankthedm, hewligan, firesnakearies and others have pointed out, had it built into the rules that characters required a specific number and type of magical items to maintain play balance, something which blows chunks if you're one of those people who prefers a lower-magic game or story, like me.
I also think there was a larger failing in 3E than simply with designs. Some respondents here have criticized the anyone-can-do-anything design mentality of 3E, and the subsequent focus on combat and combat powers above all else, with players focusing on "building" the perfect killing machine through min-maxing. Admittedly, I'm not a gamer and so what I'm thinking might have no base, but perhaps less of an emphasis on combat alone (as the CR system was so integral to the working of the game as a whole) led many players simply to focus on doing as much damage possible in a round. If there had been more emphasis on building a specific type of character in mind, whether it be a bookish, pacifistic priest (like I once generated for an online game with the DM's help), or a rogue who specializes in scouting and guerilla warfare, and designing towards that, rather than simply creating characters that could mow down the most enemies in a round, maybe it would have helped. I don't know.
I haven't played Fourth Edition, so I won't otherwise criticize it save to say that in my view they commited a cardinal sin in dispensing with the Greyhawk/Forgotten Realms conventions, as pawsplay pointed out. Dragonborn, eladrin and tieflings have no place in conventional D&D. As optional or setting-specific races, sure, but if I were a DM I'd be more than a little upset at having to retcon my world to make room for these things, when gnomes and half-orcs are sitting on the sidelines.
To be blunt, it's just not D&D without gnomes and half-orcs. That's my view, of course, but it's one I stick to.