Learn Hardcore Vietnamese: Miệng tôi đau.

Breakdown of Miệng tôi đau.

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Why does tôi come after miệng? I expected something like tôi miệng for my mouth.

In Vietnamese, possession usually works as:

noun + possessor

So:

That is why Miệng tôi đau literally looks like mouth my hurts.

Why is there no word for is in this sentence?

Vietnamese often does not use a word like English is in this kind of sentence.

Here, đau means hurt / painful / sore, and it can directly describe the subject:

So this is normal Vietnamese structure. You do not need là here.
In fact, Miệng tôi là đau would be wrong.

Is đau a verb or an adjective here?

It behaves like a stative verb or predicate adjective, depending on how you like to think about Vietnamese grammar.

In practical terms, the important point is:

So:

You do not need another verb.