Learn Polish: Bolą mnie plecy i szyja.

The verb bolą (ache) is plural for two reasons. First, the subject includes two things: your back and your neck. Second, the word for back, plecy, is a "plural only" word in Polish, similar to "scissors" in English. Even if only your back hurt, you would still use the plural verb: bolą mnie plecy.

Why does the sentence use mnie (me) instead of a Polish word for "my"?

In English, you say "my back hurts". In Polish, the natural way to talk about aches is to literally say "the back hurts me". The word mnie means "me". Polish speakers almost never use possessive words like moje (my) when talking about their own body parts, because it is already obvious whose body it is.

Is it okay to start the sentence with the pronoun: Mnie bolą plecy i szyja?

It is grammatically correct, but putting mnie at the very beginning puts strong emphasis on it. It would mean "MY back and neck ache" (as opposed to someone else's). For a regular statement, pronouns like mnie usually go in the middle of the sentence, right after the verb.

What case are the words plecy and szyja in?

They are in the nominative case, which is the basic dictionary form you learn them in. This is because the body parts are the grammatical subjects of the sentence. In Polish grammar, they are the things "doing" the aching!