No he doesn't. We don't hear reports of anyone having spoken to a burning bush, nor of booming voices from the sky. (Then again, if we did, we'd likely find those people locked up in an insane asylum.) During that time there was no technology. Things that we take for granted today, jet fighter planes as a possible example, were described as "chariots of fire that shoot flames from the sky." Meteors and solar eclipses were thought to be acts of God. If we could travel in time 2-3000 years ago and bring our Nintendo DS, we'd be hailed as gods or sorcerers. So, when looked at from a more logical, more historical-context, and less a religious point of view, it's interesting to consider what, exactly, was witnessed and called God.
As an example, read the 10-20 chapters of the Bible detailing Moses and the Ten Commandments. He actually had conversations with a very angry God and had to beg God not to kill hundreds or thousands of people. God is eternal, the beginning and the end. God is omniscient. God not only controls the world but the entire universe. "He" is the creator... of EVERYTHING. How could God be the least bit surprised at what was happening while he was on the mountain with Moses carving the tablets? How could THE God lose his temper? It doesn't make any sense that Moses was actually having a conversation with THE God about such things that God should've already known about, and not only known about but knew the outcome, and not just tomorrow's outcome but the outcome for all time.
And so, this begs the question: Who, really, was Moses speaking with? Was this, instead, Moses' experiencing what we would today call a bout of schizophrenia? Similarly, most cases where God spoke with someone involved a one-on-one conversation. We need a little more proof today, don't we? And what about those cases where many witnessed God? Read those and have fun with it. If you take the descriptions and imagine that time travel or aliens really do exist, you could easily replace "God" with some sort of modern device.
I digress, but it's still something to think about...
So, to answer your question: No, God does not talk to us like he did to Adam or Noah or Abraham.