Like someone said, the 7805 isn't the most efficient these days, but it still beats a Zener handily. For a very low current circuit, that has nearly constant current requirements, a Zener might be OK, but when used as a power supply, it's more of a "point solution". Properly designed, a Zener circuit can provide good regulation. I still use my share of linear regulators, I don't sweat the efficiency when powering circuits that only need a couple watts or less. I need about three seconds of design time for a linear regulator, a Zener circuit would take a fair amount longer, finding the required Zener current, checking the load current, calculating the proper dropping resistor and wattage, etc. etc.
But seriously, all that is needed to go into the Vin pin or power jack is rectified and filtered DC between 7 and 12 volts. I wouldn't even get too terribly carried away with the filtering. I'd go see what I had laying around, between 100µF and 470µF.