Everyone here is going to say stick to Honda/Acura ATF.
The owners manual does say that you can use Dexron III, but there is a transmission rebuilder who works for a company that rebuilds many of the transmissions for Honda/Acura cars for dealers, as in his opinion through the many years of rebuilding these tranmsissions, he says to stick to Honda/Acura ATF for Honda/Acura cars. I don't have the full details, but the Honda ATF adds a high silicone base additive that aids in lubrication and protection of some of the internal seals in the Honda transmission.
You may hear some people post stories of how their transmissions were acting a little funny during shifing until they did a drain and fill with Honda ATF.
On both my cars, 1990 Acura Legend Coupe Auto and a 1994 Acura Legend GS Sedan. They both shifted "hard" until I put in HondaATF, MAJOR difference Definately, will never go down the regular stuff again. Probably more harmful to the tranny to put the reg stuff as well judging by the feel.