An idea I find promising is delagate system. Every voter starts with one vote, which they can delagate to someone else (who can delagate to someone else, etc...). Additionally, you can reclaim or redelagate your vote at any time, or for a specific issue. This system preserves the main benifits of a republic, while allowing direct democracy.

As an added bonus, it solves the problem of rebublics like the US where a party could have 5% support and 0% representation.