This got a little complicated because my tower is just a touch out of round. That's probably due to the fact that it was pretty wet when I turned it. The result is that I had to burn the horizontal lines a bit deeper to get depth on the shallow side.
I don't trust myself with a Dremel, unless I build a jig as you mentioned. My original plan was to burn the vertical lines with a woodburner. Now I don't know if that will look bad because the vertical lines won't have the depth of the horizontal lines. I'll experiment.
I was not thinking freehand, I was thinking having it all jigged up so you could do it very precisely. I have been wanting to build some kind of jig like this myself, but haven't quite figured out the details (granted, I put the project on hold months ago). Something to hold the dremel, and some way to slide the dremel in a strait line in a particular direction. Ideally, I was thinking of using some gears, pulleys, etc. to make sure I could get steady movement, not sure if it would be required. Then I was going to use some of those really fine pointed bits to carve fine details. Simple details, strait lines and the like.
Now, I guess your tower has an angled surface...so if you did put together some kind of jig, you would need to account for that angle and make sure that advancing the dremel was in that plane... Its probably a far more complicated problem than I think it is, TBPH.