Yes, I concur with Newt. If you use some sort of herbicide:
a) It won't get all the roots.
b) It will kill all the plants in the area, including what you plant there
c) The herbicide will kill the beneficial flora and fauna in the soil that will aid in in the growth of whatever you plant and retard disease organisms. In actuallity, with long term use of anything with the suffix -CIDE you will artificially select for disease organisms.
So, definately get in there with a shovel and a Madock (pick) and dig all the bramble roots out. If some come up in the future, pull them out when they are seedlings and dispose in your compost pile.
Vinegar is definately an organic alternative to using broadbased herbicides but, don't overuse it: the acid in vinegar (very dilute) is acetic acid and will have adverse effects on the environment and your plants if overused.
Really, it's not to much work to dig things up and it can be very enjoyable and theraeutic to be out working in the garden. It's also good exercise and therefore is good for the heart.
And it's also fun to get down and dirty every once and whilte
Be sure to leave yoursefl some time to enjoy a nice cup of tea when you are done your job!