It can take a few days per roach, but many will die much quicker than that. It may be more than a week before you see a significant reduction in a big number of them -- but it does work.
The trick is to lay down a very fine layer of boric acid powder, because the roaches will walk around piles of it. Not because it's poisonous, but just like people will avoid a puddle or snow drift. (When I read that I thought it was a great description that made a lot of sense!)