Update: See the Switzerland section of the Japanese groceries in Europe page .
I have always meant to post about this but haven't gotten around to it. This is not an in-depth report with pictures and everything, but just a quick post, since Julie  asked :) If you don't live in the Zürich area go ahead and skip to other posts...
First off, I mostly cook Japanese, Chinese and Korean when it comes to Asian food. Those are the cuisines I'm most familiar with (well, for Chinese it's more like Japanese-style Chinese most of the time). So if your preference is for Thai, Malaysian, Vietnamese etc. other stores may be better.
With that in mind this is where I go the most:
There's only one totally-Japanese grocery store in Zürich, Nishi's Japan Shop, Schaffhauserstr. 120 (Tram lines 7 and 14, stop Guggachstrasse, or lines 7, 9, 10, 14 stop Milchbuch) (map ). It's small but stocks most of the basics, including rice from Japan and California. No fresh produce here except for the occasional nagaimo or something. All the fish and meat etc. are frozen, but that seems to be pretty standard for Asian groceries in Zürich. Recently started carrying real fresh tofu made in St. Gallen or Spain! (Previously they only had the hermetically sealed Morinyu type, which I dislike so much it lead me to making my own .) Also has a small selection of takeaway sushi and such.
The other store I go to a lot is Lian Hua, which has 2 branches: at Birmensdorferstrasse 94 (Tram lines 9 and 14, several buses, or any S-Bahn that stops at Bahnhof Wiedikon) (map ), and Schaffhauserstrasse 269 (Tram lines 10 and 14 to Berninaplatz). I go to the Birmensdorferstrasse one. This is mainly a Chinese store but has Thai, Korean and a small amount of Japanese items too. It has a nice fresh produce section with things like garlic chives, bitter melon, coriander with the roots on etc. (They have an 15% off everything deal on Saturdays...the sign is somewhat hidden, but they do take it off!) The only disappointment I've had here were some dried mushrooms, which were totally flavorless. On the other hand the frozen whole shrimp is a pretty good deal (cheaper and sort of fresher tasting than the ones at Migros). The tofu prices here are comparable to the ones at Nishi, but they have a wider variety. For fun they have a selection of unusual (to me anyway) soft drinks, like water chestnut and red bean.
Also, while it's not totally Asian, Barkat (which I described in depth in this post ) does carry some Asian ingredients, especially Thai. It carries more of the "Asian" in the British sense of the word ingredients, meaning Indian and Pakistani etc. Also has some Asian-type produce like sweet potatoes (not the orange American type ones, the white-on-inside ones). Great prices here! It's just a couple of blocks away from the Lian Hua at Birmensdorferstrasse, plus there's a Migros opposite, so this area is my primary everyday shopping zone. (Nishi's is too expensive to go every day...)
Finally, there is a very nice Korean grocery store, YumiHana, at Schützengasse 7 which is just off the Bahnhofstrasse, steps from the train station. (map ) Like most other Korean groceries it has a big selection of Japanese items too. Prices for Japanese items are about the same, or even sometimes a bit more expensive, than Nishi's though. I think that candy may be a bit cheaper here (like one of my weaknesses, Hi Chews...which Nishi's often is out of anyway). It has a small counter in the front of the store where you can eat some tasty Korean snacks like pajon, albeit at Züri prices.
I am happy that both Migros and Coop carry more Asian-cooking friendly produce now, like fresh ginger, green onions, chili peppers, and so on. This situation is much improved from when I first started coming to Switzerland more than a decade ago, when you had to go to the Globus food department for ginger. But, whatever you do, avoid that frozen sushi at Migros.