First off, I would highly recommend NEVER EVER EVER EVER carrying your passport around with you at all times. That document is your ONLY proof of citizenship on your trip. When you get to where you are going, find a secure place like a hotel safe or a lock box and keep it there.
Just make two photocopies: one to keep on you and one to keep at home or with a trusted party in case it gets lost or stolen. Those photocopies will facilitate replacement.
Anyways, to answer your original question:
Your debit card with the MasterCard logo will function like any MasterCard. As long as the retailer accepts it, it should work. I don't know of any retailer blacklist in any country.
Your debit card of course acts like an ATM card. If you do run into trouble, you can just use cash. Yes, its more hassle, but at least you will be able to get around in case plastic doesn't work.
Also, I would call your bank and tell them which countries you will be visiting and when you will be going. That way your ATM card and it's MasterCard function won't get shut down. But just in case, bring the collect number of your bank to call in case you do run into trouble.
It's amazing from personal experience. I was in a line at a checkout and the lady was having trouble getting her card to accept. And straight out of her mouth came "It's weird. We just came back from Mexico and used it for $300 at the Wal-Mart but then we kept having trouble with it because it would never work after that."
Basically what happened is that the card company saw a huge transaction in Mexico when the account had always been used in Nebraska and the US and their fraud department shut it down after that because they didn't call the company to tell them they were leaving the country.
Anyways, I hope that answers your question. Have a nice trip.