In Osaka -
In Hirakata-shi there's a school called Kansai Gaidai - it tends to be open weekdays and weekends (although not 24-7) and there is a postal-type ATM there that, as far as I know, takes all types of credit cards (I've defintely used Vista and Mastercard at it)

There's a postal ATM at Kyobashi station in the mall-ish section (not the Keihan Mall located at the Keihan part of the station) downstairs [it's near a supermarket] and it's open until 10pm (not sure how early it opens or how late after 10 it's open, though].

I've been able to find postal atm's in pretty much every area (including very rural) but sometimes the hours are not so great... (sometimes even worse than the lowest that patchmonkey included above, like M-F 10-14) I always make sure that I have a nice amount of cash on me before I go to Japan. (I'm not a big fan of traveler's checks as I forget about them and most Japanese stores won't accept them [e.g. you'll have to go to a bank to get them cashed])

Depending on what credit/bank company you use there also might be some large fees for using the postal ATMs. I usually get charged $5 for using a non-bank ATM and a small (luckily) out of US fee. My card does not charge me for exchanged $->yen. I'm not sure if the same fees are valid for the 7-11 ATMs as I believe it's a recent development that they accept so many different card types.

Here is some more information concerning Osaka-specific ATMs:
http://www.gaijinpot.com/bb/showthread.php?t=10228

Note: The posts are from '05 so there's a slight chance the ATMs they specify might not be around anymore.