How do we know that dinosaurs were reptilian?

you asked a surprisingly intelligent question when you used the word reptilian... which IS what they were "like" rather than saying "reptiles" which we know they werent. Its still possible that MOST species of dinosaur were more like amphibians then either reptiles or mammals. Along with a lot of the other excellent answers you got here... they all seemed to lay EGGS... a very reptilian characteristic... Bone structure and density rules out "birdlike" in most cases, and bone structures, particularly skull, rules out mammalian in most cases..... so reptilian is the "closest" set based on the largest general set of characteristics.