Great choice in camera Charlie, I got a TZ3 a couple of months back.
I had a little play last night with the macro mode after I read your post, and as far as I can tell it depended what I was trying to focus on as to what zoom level I could use. One thing I couldn't focus on when zoomed in 2x I just tried zooming out to normal and just getting closer and seemed to focus fine. Other times I needed to be a bit further away with more zoom for it to focus.
Also make sure you aren't covering the AF lamp (orange light thing) with your fingers, as I found I was doing that. Although if you were outside in daylight, that may not have come on anyway.
Also, not sure whether this actually makes a difference so I could be talking utter rubbish, but have you got the Optical Image Stabiliser on, as if its off it may be that a bit of camera shake in your hands in interfering with the focus or something

- but if it is on then which mode is it on, as the instruction book says that mode 2 is the best
