That question has an answer: they crouched like birds. A 198 million year old fossil trackway from Utah has preserved a print of a theropod dinosaur taking a break, resting with hands curled inward and knuckle down, and legs bent. Except for the forelimbs, of course, it’s very birdlike.
Here’s the section of the trace fossil they used to reconstruct the animal’s posture.
Milner ARC, Harris JD, Lockley MG, Kirkland JI, Matthews NA (2009) Bird-Like Anatomy, Posture, and Behavior Revealed by an Early Jurassic Theropod Dinosaur Resting Trace. PLoS ONE 4(3): e4591. doi:10.1371/journal.pone.0004591