I've got a few issues with that kind of thing. But my main problem would be that these people are specialist in certain areas. I'm not sure whether having a policeman having a deciding vote on what, say, Australia's economic policy is, would be really all that beneficial.
Not that what we've got not is ideal, but personally I'd rather economists deciding on economic policy, doctors deciding on medical issues, policemen deciding on crime issues etc. After all, these people are the experts...
Overall, I'm a little confused by how getting a bunch of citizens to run the country is going to solve the issue that citizens don't know what is best for them and can't run a country