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: 
The trick to be able memorize the immediate trail ahead of you and pick your footfalls several steps before you get to them.

It's the same as offroad driving. You can't see over your hood, and hanging out your window only lets you see one side of your vehicle. You have to memorize the features on the ground ahead and know where your wheels are so you can avoid those features when they disappear from view.

The same can be done while hiking. While you have your head up enjoying the scenery, make mental notes of the trail ahead, if you know where you're feet are, then you can learn to step over obstacles and avoid trip hazards without having to look down and watch your feet the whole time.
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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.