That’s why several of us have pointed at the drains - the problem could well be the rice being stuck in them, meaning the food source remains.
Considering they’re fruit flies, you could leave some radioactive material around, in hopes of inducing mutations! 
Seriously, when we had this problem, a combination of fly strips and liberal use of 409/Fantastik/Fabulous seemed to help immensely. Fabulous, in particular, contains a bleaching agent, and for some reason spraying this in areas they frequented caused them to die there in droves. Spray it high and sparingly, for maximum atomization, and then don’t wipe up the area right away, but let it sit there to poison/asphyxiate/whatever-it-does them.
Every summer, my kitchen becomes full of fruit flies. One year they were breeding in semi-moist “dried” fruit. Last summer they were breeding in the yard waste bin in my driveway, which was mostly grass cuttings, and yes, flying all the way into the house. This summer they got into some bruised potatoes in the garage. Oh my god that was bad.
I keep the (canister) vacuum in the kitchen all summer. Every time I walk in, I turn it on and vacuum up 5-10 of the buggers. Some of them get smart and head for the ceiling when I walk in. I REALLY don’t want that kind to breed! Some of them are as stupid as they ought to be, and sit quietly on the cupboard doors while I advance on them.
For the potato fiasco, I used a yellowjacket trap and baited it with fruit. Worked a charm. They did breed in there, but since they couldn’t get out it didn’t matter.
I second the vinegar suggestion, but ketchup will also work, or wine. Actually wine is really really effective.
I have not found flypaper to be very useful. It’s good for catching the tiny black flies that come from…I have no idea where…but it’s never done much to my native fruitfly population.
One day I was in a scratch-and-dent grocery store. They had just gotten in some caramel apples in plastic cases. Inside every plastic case were about fifty fruit flies, battering themselves against the plastic. My kids were clamoring for a candy apple until I showed them. They lost their appetite for some reason! :eek: I told the employees and they pulled them real quick.
Those traps from Gardener’s Supply work great, and they also sell a soapstone container to put the trap into, which makes it look pretty, and since you indicated you didn’t really want flypaper or something ugly, that might be the answer. Also pour some bleach down your drain as that will get rid of lots of them.
Great suggestions, guys! I’m going to try the dish of soapy water first, and if that doesn’t work, I’ll try the homemade insect traps or a nuclear device of some kind, whichever is easier to make. I’ll let you know how it goes.
Adam
Where do you think the bleach from your washing machine goes?