Actually, the english dub is fairly decent, but there are some key flaws that can [will] undermine the original humor of the series. The other downside is how it'll affect your general perception of the cast (as with any dub).
First, the series' "cute-factor," which you've probably witnessed a bit of in the short Movie, is central to series (at least to me). Azumanga makes cuteness seem cute, and even makes stupidity seem cute. The JPN voice cast simply sounds natural to the characters' actions, appearance, and AGE. While the english dub is decent at best, the genuine Japanese Chiyo-chan is for the most part incomparable/irreplacable, which the english cast cannot live up to. The next downer is Osaka, the girl who messed with Chiyo-chan's pigtails in the short movie; in Japanese she sounds slow/spacey/stupid but at the same time a strange childish (with a high voice) kind of cute. In english, she's got a generic mid-western american accent of a woman in her late 20s -- too deep, too mature, and i don't know how the hell they came up with that accent.
This leads me to the next point: some of the finer moments in the series are based on the premise that these girls are Japanese, who only speak *Japanese*, and have very limited knowledge of english. Various comedy bits are derived from the cultural differences, with some things specific to Japanese language/culture that shouldn't be "translated to english" in the first place -- it should be left alone. They attend an English class (shown in the short movie) at their highschool, and of course they question ("play with") aspects of the language because it's foreign to them. Now, if it was all english-dubbed, those cultural barriers get knocked down, and you've got english-speaking girls having "japanese" questions towards English -- it's just doesn't seem appropriate anymore, and would throw the comedy off (unless the script was re-written). Osaka's out-of-place mid-west accent would make things even worse. Then there's other bunches of cultural (other than language) tidbits that us westerners wouldn't be able to relate to.
Another thing the JPN cast has over the english cast: they shout and scream better -- adds more to "teh funny".
-- In short, I don't think you'd get the "complete" Azumanga Daioh experience if you saw it english-dubbed.