How to fasten boards flush to the face of square posts
You have a few options available to you, based on the material which you find. 

Provided that I understand how you want to assemble the structure you can do the following. 


Shop done Cutouts - mortice and tenon type joints
Depending on the design of the structure, I would go with this route, Just get a wood shop to saw/router various cut-outs for you, then its just an assemble job. effectively these will just ensure that there is less play. - i would use glued joints with a screw or dowel for extra strength (screw is still my preference here) 
Three anchor structures
If you are really worried about strength then just add a third support in the middle of the strucuture - you can mix it up by having a threaded bar in the middle with bolts on each side of the posts. 
Base Plates - are so simple, but effective, they improve balance and the nice thing, there are no exposed screws (since its top and bottom) once again if you are worried you can do a central dowel or threaded bar.


(see pics below)