>For the last 8 or more years I have been organically
>gardening. But in between the rows are so many weeds,
>especially dandelions that it is literally out of control.
You could look at things in a different way: some of those "weeds" can actually be considered 'volunteer garden plants' and used.
Dandelion roots can be used to make a tea with benefits for the bladder. The young leaves can be used in salads (but not old ones, just because of taste). Even the not-yet-blooming flower heads are edible.
I'll bet you also have some purslane growing between rows: it loves to take up residence in worked garden soil. It's also one of the most nutritious greens you can eat (very high in minerals and other nutrients, including being a good source of omega-3 fatty acids). Here's one site with info on it:
http://www.nutritiondata.com/facts-C00001-01c20g0.html .
And believe it or not, it tastes good! Not bitter, like nutritious greens often do. And don't take my word for it (I've been known to eat things other people wouldn't like the taste of). I tried it out on a group of people (my French Table colleagues) and they all thought it was good (though one woman refused to eat it fresh-picked from the organic garden of the woman whose house we were at: she insisted on washing it, LOL!).
Now, if there are still 'weeds' you don't want in the garden, have you tried white vinegar? Put it in a spray bottle and mist it on.
Another way of dealing with them is starving and/or them out: cover them with something (newspaper is one possibility: it is also biodegradable and will just be mulch when it degrades) so they don't get direct sun. Though if you've alredy used black plastic, that should have worked.
Getting more effective tools to get rid of them by hand could make your manual labor less unplesant. A "Claw"(
www.gardenweasel.com : many garden centers, and Walmart, also carries it) works well for disrupting the soil and the weeds (and you use it standing up). There are also specific tools (like the "weed popper" at the Garden weasel site above) used standing up that you push into the ground and turn, and they get to the root of the plants---one big problem with hand weeding is not getting the roots of long-rooted plants out, so they just return.
For on-the-ground weeding, I swear by a Korean hand hoe (lik e this one: .
http://www.gardenhardware.com/kin-shori.html quite a few natural/organic gardening catalogs and sites sell them.) As you see, they also have a long version, and I also have seen one that comes with both the short and long handle. They cut right into the dirt and are wonderful for weed wacking. I used minde to root out garlic chives from the cracks in the side walks (note to everyone: do NOT plant GARLIC chives, as they are an invasive weed. Something the people who sold the THREE plants I started with---and now fight against hundreds!--to me forgot to mention!).
ETA: Here's a site with a better price on the hand hoe (especially for the long-handled version! (I think I might actually pick myself up one of those, since I just have the short-handled version) :
http://www.kinsmangarden.com/products.asp?dept=1104 )
For sprays, "Roundup" has long been touted as less toxic to the environment, but it is not as environmentally friendly as they want you to believe: if it gets into a water supply, it can kill fish and other critters.
>> In the meantime I'm clearing out weeds by
>hands.
That's what I do!
Just think of it as an upper body exercise (and get the right tools to make it easier and faster).