Are humans adapted for a savannah environment, as opposed to the forest?

We always depict 'natural' human beings as living in tribes in the jungle, but I'm not sure that's true. I read somewhere we spread out onto the savannah as soon as we could walk. I also feel like the fear of the woods is the most primitive instinct I have. The woods at night is the scariest place I can imagine being, whereas the flat, open savannah seems like a paradise (even if in my mind I know there's lions there). I feel like this is the opposite of the way a monkey would feel. It would look to the woods for the safety since it is actually adapted for the woods.

Which is more likely? How long ago did we emerge from the woods, and will evolution have operated on us fast enough since then?