Spread a tarp under each raspberry bush to collect clippings. Trim back the bushes with sharp pruning shears, but do not cut them back to the ground. You want to cut excess vines to see where in the ground each plant is growing.
Locate the spot where each bush grows, and cut vines and shoots using sharp pruning shears, leaving only the stump behind. Start from one side of the bush and work your way to the other side to remove the growth completely. Alternatively, dilute a glyphosate herbicide with water in a pump sprayer and apply it to the growth on the bush until saturated. Position the nozzle of the sprayer over the stems and foliage and spray them with the formula. The herbicide kills the growth in four to seven days, making it easier to cut the plant down.
Dig around the perimeter of the bush to pull the roots out of the soil. Using a trowel or gardening shovel, dig to a depth of 8 inches or more to completely expose the roots. Angle the tip of the shovel under the root ball and pry it out of the soil. Sever stubborn roots that grasp the root ball in the soil with a hatchet.
Collect the root ball, roots and clippings in the tarp. Remove broken roots from the planting hole and add these to the tarp. Do not add the clippings to your compost bin or pile, but discard them to prevent chances of regrowth.
Spray a weed killer into the planting site to kill raspberry roots that remain. Alternatively, spread a barrier such as a black plastic sheet over the area to prevent future growth. Monitor the area for the next couple of weeks and spot-treat raspberry sprouts that emerge from the soil.