Hi Androu;I can't tell you what the position is in italy, but I can tell you what the reverse position would be, ie using italian money in England.
First off, I'd suggest that you're probably not moving that much actual, physical money around within a country, let alone between countries.
Actual money is heavy, inconvenient and an invitation to robbery. If you're a merchant or noble, you aren't going to do this if you can avoid it.
Much better to get a letter of credit by depositing your cash with one of the italian banks before you leave, and just pick up what cash you need at your destination. Mechanisms for this are well-developed by your tme period (ESPECIALLY in italy), and well-attested.
For small amounts - if you're a soldier with a few coins, for example - there's still no real problem spending it in another country.
Most medieval coins have intrinsic value, being made of bullion, albeit diluted to a greater or lesser extent, and can therefore be spent anywhere. Indeed, England actually suffers from coin shortages in this period, and large amounts of continental coinage comes into use just to keep the economy liquid.
Any merchant of standing is going to possess a coin balance and a good knowledge of the bullion content of the most common foreign coins.
You'll have more of a problem with small traders or outside towns, where people will be less sure how much a foreign coin is worth. This won't stop them taking your money, but they'll probably err on the conservative side when converting, so you'll effectively pay a hefty markup for paying in "Furrin money" rather than good english silver.
Finally, it's worth remembering that there's no such place as "Italy" until 1870. In 1360, italy is a mess of competing city states (Florence, Milan, venice etc), all issuing their own money. If anything, I'd suggest that a place where the money changes every hundred klicks or so is going to be even more used to working with foreign currencies than a big, unified kingdom like england.
Hope that helps
Neil