What a great idea....One thing with tyre's is that they leach Zinc. Zinc can be toxic to plants but spuds actually like it so you should be OK.
My preferred method of doing the sleeper bed is to lay 100 x 200mm sleepers on the flat, alternate the corner overlap and hammer a 150mm landscape nail through the corner pinning them together.
Its a lot less work than tripper method with the posts, but it takes twice as much timber...and I'm lazy.