Thanks to everyone for all of your replies.

Just so that everyone knows, the objects I'm working aren't just a few. There are tons and tons of objects, so it's not something that can reasonably be done by going one by one. It's not a "centipede" of objects, so to speak, that directly connect into each other, such as a leg to an ankle, an ankle to a foot, etc. They're all scattered about. I'll attach a picture to make it more clear.

As for the whole boolean modifier situation, if someone could explain exactly how Cybermonk's suggestion would work, I would really appreciate it. I'm a little confused on exactly how that's supposed to work.