Learn Italian: Controlliamo tutti gli orari dei treni.

Yes, but you would have to change the noun to singular! Ogni (every/each) is always followed by a singular noun without an article. So you would say ogni orario dei treni. Because this sentence uses the plural noun orari, we must use tutti gli (all the).

Why is there a gli after tutti?

In Italian, when you use tutti (all) with a noun, you almost always need the definite article between them. Since orari is masculine plural and starts with a vowel, it takes the article gli, forming tutti gli orari (all the schedules).

Why does it say orari dei treni instead of just translating "train schedules" in that order?

Italian doesn't stack nouns together like English does. To say "train schedules", you literally have to say "schedules of the trains". This is done using the preposition di (of) combined with the plural article i (the) to make dei (of the), resulting in orari dei treni.

Why do we use the article gli for orari but i (hidden in dei) for treni?