How do I kill wild razzberry bushes in my garden without killing the rest of my flowers?
Raspberries are a rhizome, meaning they sprout from the root. Remove the root, you remove the bush. The problem is that even a microscopic piece left behind will sprout anew. You are wise to be leery of using herbicides such as Round Up because it will not only kill whatever it touches, it will be drawn down into the root system and leave residue behind in the soil (contrary to the advertising of the product ). That residue will affect any plants in the vicinity of the offending plant location.
To kill such a beast, without harm to surrounding vegetation will require either devout attention by the gardener to pull sprouts as they appear - eventually killing the roots, or the use of a torch to singe the plant and make it bleed. A weed torch is used to burn the plant just enough that it can't heal itself and it basically bleeds to death. If you burn the greenery to the ground, all you've done is encouraged it to resprout.
Click this link to see a review of a torch that I purchased. There are pro's and con's to the different units available and a general instruction in the use and intent of a torch.