Do I need a visa to visit Greece, Romania and the Czech Republic? I have both UK and Australian passports.
In short: no, you do not need a visa to visit those countries.
Greece, Romania, and the Czech Republic are all EU members. At the time of writing, holding a UK passport means the right to free movement within the EU, enabling you to visit (or even reside long-term in) other EU countries without a visa. It is currently unclear which requirements a UK passport-holder will face for travel within the EU after Brexit, but as you're also an Australian passport holder you don't need to worry about that (see next paragraph).
Greece and the Czech Republic, as well as being in the EU, are also members of the Schengen zone. Australian passport holders can enter the Schengen zone visa-free for stays of up to 90 days. Romania is not a member of the Schengen zone, but also allows Australian citizens visa-free entry.