For such a technologically advanced country, Japan can be surprisingly backwards at times, and dealing with ATMs and credit cards is one of those times. I’ve been to Africa and Southeast Asia and had no trouble using my American ATM card but I can’t use it in most places in Japan.
There are some ATMs (see below) where you can use a non-Japanese ATM card. Hours might vary (in Japan, ATMs at traditional banks generally have operating hours), but general guidelines are below.
Yucho Ginkou (the Post Office). They take foreign ATM cards and have the ATMs in English as well. Operating hours are a bit short (smaller post offices won’t have their ATMs open at all on Sunday and will close at around noon on Saturday), but the post office is everywhere (it’s called “yuubin kyoku” in Japanese). Unfortunately there is no English search page, but searching for “post office” in Google Maps actually brings up the Japanese post offices.
7/11 (the convenience store). They take foreign ATM cards and have multi-lingual ATMs as well (English, Chinese, Korean and Portuguese, I believe). They’re open all day (unless the shop closes for some reason), but you can only take out cash in increments of 10,000 yen (around $100US). In addition to being in the convenience store, they can also be found in train stations, malls and shops. 7 Bank (who operates 7/11 ATMs) has an English search page here.
Shinsei Bank and Citibank. These are the two rarest options, but options nonetheless. They both take foreign ATM cards and have English ATMs. They’re open all day and night as well, unlike Japanese banks. Shinsei Bank branches can be found here. Citibank branches can be found here (note: this is a PDF file).