In addition to Richard's quote from J.K. Rowling, Fantastic Beasts and Where to Find Them simply describes Thestrals as "black, possessed of the power of invisibility, and considered unlucky by many wizards." Yes, the Ministry at one point classified Thestrals as "dangerous"; however, the imperfections of the Ministry could fill another seven novels! Anyhow, when Umbridge points out the Ministry's classification of Thestrals as "dangerous", Hagrid attempts to clarify:
‘Thestrals aren’ dangerous! All righ’, they might take a bite outta yeh if yeh really annoy them –’
‘Shows ... signs ... of ... pleasure ... at ... idea ... of ... violence,’ muttered Umbridge, scribbling on her clipboard again.
‘No – come on!’ said Hagrid, looking a little anxious now. ‘I mean, a dog’ll bite if yeh bait it, won’ it – but Thestrals have jus’ got a bad reputation because o’ the death thing – people used ter think they were bad omens, didn’ they? Jus’ didn’ understand, did
Order of the Phoenix - page 396 - Chapter twenty-one, The Eye of the Snake - Bloomsbury
There is nothing I know of in canon that suggests Thestrals are typically mad, bloodthirsty carnivores, who run amok through the forests of Britain, looking for random humans to victimize. J.K. Rowling did not characterize Thestrals as typically dangerous to humans because Thestrals are her creation and she gets to choose their demeanor. However, within those confines, I think it's safe to assume that, like any being, there are circumstances under which a Thestral might get aggressive toward humans. For example, if a human taunted, deliberately injured, or withheld food from a Thestral, the Thestral could conceivably attack. Hagrid says as much in Order of the Phoenix, as noted above. In canon, the Thestrals are depicted as shy, docile, and tame. See the picture below.
Harry and Luna Feeding the Thestrals by Rob Bliss, from Harry Potter: The Creature Vault by Jody Revenson
In Order of the Phoenix, in chapter seventeen, Professor Grubbly-Plank does confirm to Harry that Thestrals occasionally eat birds.
Another consideration is the possibility that Thestrals are carrion beasts, who forage instead of hunt.