Learn German: Der Mann rannte aus dem Restaurant, ohne dass der Kellner die Rechnung brachte. - The man ran out of the restaurant without the waiter bringing the bill.

You use ohne ... zu when both parts of the sentence share the same subject. Here, the subjects are different: the man (der Mann) is running, but the waiter (der Kellner) is the one not bringing the bill. When the subjects are different, you must use ohne dass.

Does ohne dass change the word order?

Yes. ohne dass is a subordinating conjunction, which means it kicks the conjugated verb to the very end of its clause. That is why brachte (brought) is the last word in the sentence.

Why is it aus dem Restaurant if he is moving out of it? Doesn't movement mean we should use the accusative case?

The rule about movement versus location only applies to two-way prepositions (like in or auf). The preposition aus (out of) is a strict dative preposition. It always takes the dative case, changing the neuter das Restaurant to dem Restaurant, regardless of movement.

What tense are rannte and brachte?

They are both in the simple past tense (Präteritum). rannte is the past form of rennen (to run), and brachte is the past form of bringen (to bring). Both are mixed verbs, meaning they take regular past endings but change their stem vowels in the past tense.