I'll pile on about bank employees (not necessarily officers) being ignorant.
The BECU person who told you that is WRONG, WRONG, WRONG! With the few exceptions already listed, your strip and signature card will work fine in Italy. My BECU Visa did in October and November in Fiumicino and Milan, so you have nothing to worry about there.
My BECU MC Debit card also worked all over the place to withdraw money from ATMs in October and November, in Italy and Greece.
We did not charge a single meal on our 6-week trip to Istanbul, Greece and Italy. We paid cash for all of them. We did use the BECU Visa to charge a few hotel bills, but mostly we paid cash for those that we did not pay in advance online.
My husband used his BECU Visa to buy me a very nice gift which was shipped home and a surprise when I got it. I used mine to buy a rather expensive rug in Istanbul, and because of the price, we used the merchant's phone to call to verify the charge. BECU is hyper-vigilant when it comes to potential fraud, so in the midst of all your confusion, be sure that you tell them exactly where you might use your credit or debit cards. Likewise, do that for any other cards you might take along. We did have to call them from Italy when we got stuck and realized that we were going to be there past the date I said we would return. Next trip, I'm going to put a little more slosh in our standby schedule.
I got a new BECU Visa in January. I was hoping it would be a chip and pin, but BECU will not be doing that until the fall, or so I was told. And even then, they will not replace the old cards until they are due for renewal (or unless they are forced to, is my guess).
Go! Have fun! Spend Euros! Eat well! And enjoy your trip.
Edit: Call this number to get the straight answers from BECU for both your CC and DC. It gets you past most of the "for XX press XX" idiocy. Number 206-439-5700-1-7457. Follow the prompt and that will take you directly to the extension for the people who know what they are doing.