One of the characteristics of the Thailand is that they eat sitting on cushions on the floor. Most authentic Thai restaurants also use this system, although they have normal tables in the Ruen. The restaurant is in the heart of Ayutthaya, across from the temple Ratchaburana, and is a perfect place to relax with a drink and AC in between visits to temples. You can also order food off the varied, cheap menu (something not easy to find in touristy Ayutthaya). It has good, authentic Thai dishes starting at 4€.
When we were in Ayutthaya, our tuk-tuk driver took us to this restaurant. We assumed he was getting some kind of commission for it, but the truth is that we ate really well and it was reasonably priced. It was tourist-oriented though, and there were lots of foreigners there. All in all, it was pretty good.