I have read a myriad of articles about the evils of using Herbicides and in general I am in complete agreement that the less we can use these products the better. In my garden I am 100% organic and will continue to be as long as I have anything to say about it.
That being said, I told you I would give you several options to get rid of bindweed. One of those options is Herbicides. Specifically products like Roundup - which will kill pretty much any plant that it gets on; and 2,4-D (2,4-Dichlorophenoxyacetic acid) which is a systemic herbicide that kills broadleaf plants. I have used both of these products in the past and had varying results on bindweed. The Roundup killed the bindweed after several treatments, but it sprang back up in the same location the next spring. The 2,4-D killed the bindweed in my lawn, but once again it reappeared the following spring.
So, in the spirit of the Organic Gardener my recommendation is that if you can try to use organic methods instead of resorting to herbicides you will be doing a great service to the earth, your yard and your pocket book (pulling weeds by hand is free). And, if you compost your grass you won't want that stuff in your compost heap either.