Which D-SLR Camera: Nikon or Canon?

I'm a serious amateur photographer. I currently have a D5000 from Nikon with only the kit lens and an SB-400 Flash. I really like it, but I feel like I'm a bit limited with its "entry-level" feature set. I have some money saved up and I'm looking to get a mid-range to professional level D-SLR. I would like to get something like the Nikon D7000, D700, or Canon EOS 5D Mark II. Which one do you think I should get? Is there any specific reason to go with either brand? I will be photographing mostly travel, sports, and nature. Also, any suggestions for lenses?

I'm no pro, but I studied with a pro for quite a while. After talking with him and other photographers, and after doing some research myself, I would definitely suggest that you stick with Nikon. The photos that you get out of a Nikon tend to be a lot more crisp and sharp than those you get from a Canon (or other brands, for that matter.
In the end, it seems the two brands are pretty closely matched in the industry, but this point sealed the deal for me.

What feature set does the D5000 not have that a professional camera have?

More professional cameras actually lose features, like the ones that tell the camera what to do when you have sports and whatnot. Those features are nonexistant. Don't expect the quality of your pictures to be better once you get a new camera. Only get a new camera if there's something limiting you from doing what you want to do with your D5000.

Everything the Nikon D3x can do, the D5000 can do, for the most part. The only real difference, is size, weight, sensor size, and megapixels.

The thing you should be looking at getting new lenses instead of a new camera. Lenses make much more of a difference than the camera. Look at getting a super wide angle, or telephoto, or a portrait lens, such as the Nikkor 50mm f1.4 to expand your shooting ranges and potentials.

again, you do NOT need a new camera.

I'd examine why you feel limited. What is that combo not doing for you? All the cameras you mention are fine cameras.