I don't think it's all that bad to put your graph on ordinary everyday axes as long as the students know that the order is more or less irrelevant.
If you are happy to break out of the page, I recommend drawing your graph on a piece of paper and rolling it up to make a cylinder. Then at least one of the axes represents the cyclical nature of the field.
If you want to have the graph on a torus on a screen, it may be better to use technology.
I've used Geogebra to make a workbook of various graphs, including cartesian graphs, cylindrical graphs and input-output graphs for $\mathbb{Z}_7$.
The cartesian graphs looks like the ordinary graphs students would be familiar with:

The x-axis and y-axis can both be moved so that the labels cycle around, giving you the feel of the toroidal shape. UPDATE: There's now an activity that allows you to choose the order of the ring, from $\mathbb{Z}_3$ to $\mathbb{Z}_{30}$.
The cylindrical graphs are like the cartesian one, only the x-axis has been rolled up to make a circle:

The circular axis at the bottom can be rotated, and the vertical axis (the separate circles) can be slid up and down so the values cycle around. (It looks much better in motion than statically.)
Finally, the input-output graph has a domain at the top and a codomain at the bottom with arrows to show where each point goes under the function.

You can spin the top and bottom independently or together, and change the perspective.