Question: Every time I go walking or hiking, I generally spend most of my time looking down to make sure I don't step on a snake or hole or dog poop. I don't do it consciously, it just happens. Obviously, I miss out on a lot of birds and small animals up in the trees and I'd like to see how other people cope with the urge to avoid obstacles and just enjoy their surroundings.

How can I go walking and hiking without constantly looking down?

Answer: 
I find this is something I do when I can't see much of the scene in front of me at once. This is normally because I'm not far from the person in front of me. Therefore, my advice is: walk in front. Or, leave a big gap between you and the person in front of you. This is perhaps somewhat antisocial, but personally I quite like to walk alone about half the time. It lets me focus on my surroundings a bit more.
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> How can I go walking and hiking without constantly looking down?
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When I am walking in a city such as NYC or Las Vegas, I do not look down. I am looking right and left for cars, people and dark alleys. But I think you are correct when in the wild to look down more than other places. As Cody Lundin says foot placement is very important. When stepping on rocks, up and over logs, in and out of mudholes and through high grass, you just have to know where you are placing your feet. There is only one way to know, you travel slowly and deliberately and do not place your feet in any spot that you have not looked at first. 

Sure, that means cutting your speed down but I do not see any other way. We hike in pairs so that the front guy checks the ground out and generally only looks 15 to 20 ft ahead. He is clearing the way for the guy behind him who scans the trees and listens, getting an overview. Depending on where you are of course we do not relax and smell the coffee until we reach the destination. If you want to look around and enjoy nature, stop! We never do that moving.