I have been considering this problem for some time now, if only for British politics.
One slight problem is that people are dumb. Senators and MPs act not only as a geographical proxy, but as a mediator for stupidity.
Take for example the MMR vaccine scare. After this event, the majority of the public were outright scared of this vaccine. It was the duty of the politicians to educate themselves on this issue, and think about things rationally.
In addition, educating oneself on a matter takes time. Most people will not be able to find enough time to learn about these issues.
This could be partially solved by a few methods.
Firstly, only a randomly selected subset of people could be permitted to vote on each issue. This would give people the opportunity to learn about the topic at hand, and make a rational informed decision.
Still, people can be swayed by either effects, and most people are very capable of making bad decisions, even if they know all of the information.
This could be solved by having people take a "Voting Exam". In the same way that a person driving a car, is likely to do harm if they are not able to control the vehicle, so could a person voting irrationally do harm. I suppose that the voting exam would consist of testing the subject's critical thinking skills and their ability to comprehend and utilize new information.