Depending on the type of weed, if you can't get all the root it may grow back no matter what. A thick bed of mulch can help. If it is coming up from seeds, spreading a corn germ pre-emergent on the beds will prevent their germinating for awhile, and it breaks down to feed the soil. If you use it, you will need to prestart your seedlings as the pre-emergent will prevent your veggies from germinating too.
Depending on the weed type, a vinegar spray on young growth could be effective. If you want to use Roundup, you could use a foam paintbrush to apply it just to the weed leaves and not have to spray your whole garden bed. Use any herbicide when the offending plants are small and on a breeze-free sunny morning.
A few years ago, shortly after I had built a new fence, a new neighbor moved in next door. He relandscaped his entire yard, adding a raised bed in one corner. Having no idea that he was using my fence as the rear retaining wall, the soil contact rotted out the fence and created a freeway for termites to finish off the entire fenceline.
You can get a wealth of info here:
http://organicgardening.com/