Actually, @Kropotkin, that’s not entirely fair to the Corvidae family, specifically Corvus (Ravens, crows, etc). They’re highly intelligent, actually, are tool users, learn through play, and are capable of understanding human language (Ravens can be trained, for instance). I think that, given a few more million years and the proper selective pressure, we could see a Corvidae of human intelligence.
The selective pressure is the key, though, and makes these questions really hard to answer (fun to speculate on, though). In keeping with @LuckyGuy‘s predictions, I would have expected mammals to stay mostly small prey species, though I’m not sure we would have ever evolved such high intelligence. Larger brains are extremely expensive pieces of equipment to use and evolve. With large ecological niches opened up by the deaths of the large predators, we were able to evolve to capitalize on the mammalian adaptability and become predators in our own right, then were able to use larger brains for a predatory as well as adaptability advantage. With large predators still existing, we may never had had the ability to evolve in this way, and smaller mammals with decent but still modest brainpower may have been all we got.
Now, I do think (going of my Corvidae rambling above) that dinosaurs may have been capable of evolving intelligence, given enough time. Exactly like mammals? Probably not. But the various raptors were quite possibly intelligent as well as social, such as Velociraptors and Deinonychus. Also, the advent of efficient flying reptiles, already on their way in the form of feathered raptors, may have given them enough of a competitive advantage to allow a large degree of diversification and evolution of higher intelligence.
So, yeah. I doubt mammals would have come to the fore without the extinction event, but I think it still possible we would have had higher intelligence of some kind. Raptors that can use door handles, for instance.