From all the legalism in Islam, I like to think of Assyrian Baghdad would have belly dancers and Persian style(ish) carpets and hookahs and uniquely built churches that resemble mosques and more peaceful before the introduction of Islam in modern day Iraq, I would like to think something like Aladdin's agrabah type world but minus Islam and just have Jews and oriental orthodox Assyrian Christians.
Iraq was actually a region of the Sassanid Persian Empire, and was only rarely held by the Roman Empire. The Assyrians Empire was long gone (collapsed circa the 7th Century BC when it was conquered by the Babylonians). Assyrians, however, are an ethnic group on the Euphrates that were Nestorian Christians.
Baghdad would be build by the Abbassid Caliphate, so it did not exist. What did exist was Ctesiphon, which was the Capital of the Zoroastrian Sassanid Empire. The city would have been well fortified and would have been home to a number of world religions, Nestorian Christianity, Zoroastrianism, Manichaeism, and a few other Gnostic Sects (Mandaeanism, and Sabianism to name a few). Churches would have likely taken on a Roman basilica form.