This is something I've been thinking about recently.
Let's say you had an undeciphered language on your hands. If the grammar was identical, or at least similar, to your own native language (or at least a known language), but the vocabulary was not, would that make it easier to decipher than a language that was alien in both vocabulary and grammar?
I've heard of codes that involved simply replacing words, but those are mostly used by children. I have heard about the story of the US using a code that simply involved replacing words with the expressions that the Navajo used in their native language (such as referring to a tank as a 'turtle'). But its possible that they used some sort of further decryption after that. Such as one of those ciphers you need a specialized machine to encrypt and decrypt the message. I can't remember what they're called, but they looked like typewriters.
Personally, I suspect some aspects of grammar may make decipherament easier. If a language marked things like case, tense, definiteness, and/or number, I don't think it would be that hard to notice affixes that were used in virtually every sentence, possibly multiple times (such as how its common for English sentences to use the definite article two or more times). I mean, people have noticed what may be affixes in the Voynich manuscript, though they have no way of knowing what they mean. Personally I feel like these 'affixes' appear too frequently. I mean, they appear 3 to 4 times at least in every single sentence! Unless they're marking part of speech, I don't see any affix being used that frequently.