When the output goes high (+12V) I want a red LED to turn on, but when the output goes low, I want a green LED to turn on. Here's the circuit I came up with: (see attachment)

The onlyl downside: with the Output in Hi-Z, the circuit is drawing 6mA. Anybody know of something better?

 

What you seem to be making is a "logic probe". There are lots of designs - http://www.electronics-project-design.com/LogicProbe.html
http://www.circuit-projects.com/test-equipment/cmos-based-logic-probe-circuit.html
http://www.circuit-projects.com/test-equipment/cmos-based-logic-probe-circuit.html