Currently, there is no automatic way to have a table with alternating row fills AND alternating column fills; the table settings will only allow you to set one or the other, and you must manually change the colors of the rows (if you set columns to alternate in the style) or columns (if you set rows to alternate in the style) from there.

It would be nice if we could set BOTH columns and rows to have alternate fills. For example:

You have a table with five columns, ten rows. You tell every other column to be 100% Magenta. You then set every other row to alternate as well between 100% tint and 50% tint, using the column color by default.

Now you have a table where every other column is magenta, and the rows alternate between 100% magenta and 50% magenta.