The issue here is that you have added all your banners, divided between clients and categories and you're looking for a solution which doesn't involve installing a 3rd party applicatoin.

 Unfortunately the Joomla banner module only supports alternating between banners of either the same client OR the same category.  The setting is in your Joomla > Module Manager > Banner... What I do is I don't select a client but instead just select a category.  Set it to only display 1 banner at a time.  

If you want other categories to display, consider setting categories as vertical, horizontal and square banners.  This means that you can have 3 modules each displaying banners from each category and position these as appropriate.