Yup, like the others said, oil is the way to go! Grease will just make problems.
My one son decided to help me out with my Logan lathe shortly after I got it, and he applied some grease to the lead screw using a rag. It was a light coating, but it still held little metal chips. I finally cleaned the entire lead screw with degreaser, then oiled it lightly with 30 wt non-detergent motor oil (that which is recommended by Logan) and things are good again. Way oil would also work, but is a bit thicker than motor oil.