What spices were popular in Italy during the 1400s, 1500s, and 1600s?

Not rosemary — that’s silly. Rosemary grows anywhere, you don’t have to risk your life to get it.

I’d say pepper of various types or basically any kind of oriental spice. These either had to come via a long sea voyage around the bottom of Africa, or a long caravan voyage across the Samarkand Desert (basically Mongolia). Both of these were very hazardous and therefore very profitable journeys.

Pepper was shipped in bulk and cost pound for pound far more than gold.

See the link below for many details.

“The term ‘spice’ actually covers a host of sins, including dyestuffs and a wide range of drugs and apothecary materials. But I am confining this discussion to the more traditional usage and on the screen I have indicated the names of the chief spices, with their Indian names… Far and away the most important was pepper, which was always shipped as a large bulk commodity; followed by cinnamon, ginger, cloves. I have omitted two lesser spices used in medieval Europe but only rarely today: cubeb and galingale (the latter being close to ginger).”