Greek word κλινη (Klinê):
tri-, "three", + κλίνη, klinē, a sort of "couch" or rather chaise longue
They used to eat, like in Peplum movies, sitting on a"chaise longue" not on a chair.
How it was placed:
Note, the "normal" position to eat, during more than 1000 years, was lying down. It changed with the new christian religion, as eating lying down was probably associated with debauchery, and a lack of dignity.
Under Napoleon, eating lying down came back, with "méridienne". (They are not "chaise longues", as "chaise longue" in modern French is a beach sun longer)
Note that this was most likely only true amongst the aristocracy and nobles, but we have a lack of any evidence of other practices. But then if the plebeians ate sitting down with food in their lap, they wouldn't leave much evidence of it. And frescoes almost always depicted the life of the aristocracy.
"Dining room" is a bit of a misnomer with "triclinium" since that's really just an arrangement of three (tri) lounge chairs (clinium) for the purpose of eating. "Cenaculum" is an actual room per se.
Other comments on this page provide details as to what a triclinium is.