I'm probably going to bring down a storm of people saying how wrong I am, but Japanese is somewhat like that.
I say somewhat because intonation is used in the language, and quite a bit, but its not needed for understanding. A good example is turning a sentence into a question. I hear all the time people using a rising lilt at the end of a sentence to convey uncertainty, that is, that they're asking a question. But they don't have to. They could just use the verbal modifier that adds a question element at the end of a sentence.
If someone says "genki da?" you just hear the "?" at the end from the intonation in their voice, asking if you're doing well. But if they say "genki da ka" you would know it was a question because the "ka" means, in English, "?". It's a word indicating that the rest of the sentence is a question. There are all sorts or particles and conjugations for things like that.
Japanese isn't a language that doesn't use intonations, since people are people, but I think that it could use no intonations and, without changing the language at all, still be as useful a language as it is now.