I've been on a sortition kick for some time now and am a big proponent, but simply choosing congressmen at random would never work. I believe that with subcommittees and some sort of filtering mechanism, it could work very well. The filtering mechanism is the trickiest part though. So far I haven't come up with anything better than a qualification exam, which has some obvious flaws.
The way I see it working is like this: You have a central committee that is like the legislature, but they don't draft bills. They only debate them and vote on them. This committee is made up of at least 40% unfiltered complete randoms. The term could be anywhere from 6 months to 4 years; the position being non-compulsory and well compensated. The central committee also selects someone to serve as the head of state, who will act kind of like the SoS and be commander in chief of the military. This person could be replaced with a vote of no confidence like a PM.
The actual legislation would be written by filtered subcommittees. For example, you could have an infrastructure committee that has two lawyers, 2-4 engineers, two economists and two social scientists. These would be selected at random from a pool of people who have passed some sort of qualification exam. The exam would be very difficult, but people would be highly encouraged to take it to maximize the size of the pools. Of course study materials and services should also be provided by the state to make sure that everyone has approximately equal access.
It's not an ideal system, and there are some flaws, but I believe it would work very well. Having committees of experts crafting legislation would limit the influence of lobbyists. You could also potentially have some sort of petition system to keep effective leaders around for a second term.