I'm having good results with Bt ( Bacillus thuringiensis ), but I don't think aphids are listed on the label as being controlled by it.
The main problem I'm solving with Bt is cabbage loopers.
Wikipedia says: Cry toxins have specific activities against insect species of the orders Lepidoptera (moths and butterflies), Diptera (flies and mosquitoes), Coleoptera (beetles), hymenoptera (wasps, bees, ants and sawflies) and nematodes.
Aphids are order Hemiptera, so it doesn't look like Bt will work on them.
I have heard of using tomato leaf or garlic oil spray to control aphids, but haven't had to try either.
See this page for recipes , or google around.
gardening.stackexchange.com/users/51/bstpierre
I have found that a soap solution works really well - and it is safe!
If used promptly, it is also effective in ridding broad beans of blackfly.
gardening.stackexchange.com/users/136/Mancuniensis
I've used neem tree oil mixed with organic dish soap on edible container grown fruits and veggies and have always had good luck.
Neem oil is edible but you'd probably want to wash your fruits/vegetables before eating because the oil is quite bitter.
gardening.stackexchange.com/users/13/somas1
I use pyretheum.
Is it organic enough for your needs?
Depends.
Back in the UK "pyretheum" is the 'natural' variant made from chrysanthenums (and the one I used to go for); and "pyrethins" are the synthetic equivalent.
But here in the US there does not appear to be any distinction and everything is labeled "pyrethin" plus various claims to being a natural pesticide.
Pyretheums are particularly good for delicate plants (I've used them on carnivorous plants - don't laugh!, but hear they're also good for orchids) and veg/fruit.
Even though they break down quickly, you would still wash the fruit of course.
My wife (an ecology Univ.
Prof.) usually swears by soap solution for aphids, although I haven't seen her use it.
gardening.stackexchange.com/users/69/winwaed
My housemate suggests using black tea (cooled!) and spraying it on the leaves that have aphids on.
I cannot find a cite for this!
However, I did find an Aphid repellent tea which looks interesting.
Also, there's some interesting information on companion plants that will help against aphids, here .
gardening.stackexchange.com/users/317/Suzanne Hillman
While not the pesticide you asked for, you might consider a pest management plan that entices ladybugs to settle in and combat your aphids.
Even "organic" pesticides can be high profile pesticides and kill many beneficial organisms.
gardening.stackexchange.com/users/697/Erik Olson
I use neem seed oil effectively.
From the wikipedia article (emphasis not in original): Formulations made of neem oil also find wide usage as a bio-pesticide for organic farming, as it repels a wide variety of pests including the mealy bug, beet armyworm, aphids , the cabbage worm , thrips, whiteflies, mites, fungus gnats, beetles, moth larvae, mushroom flies, leafminers, caterpillars, locust, nematodes and the Japanese beetle.
Neem oil is not known to be harmful to mammals, birds, earthworms or some beneficial insects such as butterflies, honeybees and ladybugs if it is not concentrated directly into their area of habitat or on their food source.
gardening.stackexchange.com/users/848/divine morny
In Indonesia we use the product "Samro biocide", which is a multipurpose, organic pesticide for nematodes, fungi, bacteria and insects, with special funcion as growth recovery.
This product is basically made from a herb extract and is modified from herbal medicine with growth recovery, because it contains antibacterial and antimicrobial substances including antibiotics and immunomodulators.
The materials are more than 20 original secret Java herbs extracts (from nimbi, java turmeric, small gooseberry, ginger, black turmeric, bitter grape etc.) obtained by taking liquid secondary metabolites of certain parts.
gardening.stackexchange.com/users/707/Lesto Kusumo
You can also plant garlic/onion in your pots or in proximity of your plants in case of garden and see aphids disappear in some time...
this has worked for me for over a year now, I plant garlic with every flowering plant and aphids have kept away....
washing your plant regularly in jet stream also helps them keep away.
gardening.stackexchange.com/users/13562/mbaxi
If you are into biological control look at some parasitic wasps, or if you can access, we use an "organic" chemical named Endeavor.
It works phenomenally against aphids!
Usually with one app, but you can double app after 3 days if needed.
gardening.stackexchange.com/users/26665/Cara
I have a vegetable garden that bugs just love. Since my family is going to eat these vegetables (and I'd prefer that the bugs do not), what organic pesticide is most effective? The main culprits are aphids and cabbage loopers (or inch worms) though any pesticide that targets critters that eat leafy vegetables would be helpful.
What is an effective organic pesticide to use against aphids or cabbage loopers?
gardening.stackexchange.com
<pesticide><organic>
gardening.stackexchange.com/questions/76
gardening.stackexchange.com/users/79/Michael Todd
You can try a soap solution, possibly mixed with neem oil. Try planting onion or garlic in your pots to get rid of aphids. Soap solution works really well and it is safe. Neem tree oil is effective, especially if mixed with organic dish soap.  You can use tomato leaf or garlic oil spray to control aphids or you can also plant garlic or onion in your pots or near your plants. Pyretheums are particularly good for delicate plants. 
Back tea, cooled and sprayed on the leaves that have aphids on is another suggestion.   You could also consider a pest management plan that entices ladybugs to settle in and combat your aphids.
Soap solution works really well and it is safe! Neem tree oil is effective, especially if mixed with organic dish soap.   You can use tomato leaf or garlic oil spray to control aphids or you can also plant garlic or onion in your pots or near your plants. Pyretheums are particularly good for delicate plants. Back tea, cooled and sprayed on the leaves that have aphids on is another suggestion.    You could also consider a pest management plan that entices ladybugs to settle in and combat your aphids. S0: In Indonesia we use the product "Samro biocide", which is a multipurpose, organic pesticide for nematodes, fungi, bacteria and insects, with special funcion as growth recovery.