As a banker I offer the following simple ideas about paying when travelling. As an aside, I look forward to the day when I "get rich off all these fees" so I can quit working and dive full time. That hasn't happened yet, but I digress...
Cash
- highest immediate risk because if it's gone, it's gone
- more tempting for unsophisticated criminals to try to take from you
- offers the greatest protection of your checking account and credit report
- "no" conversion rate when USD is accepted, unless the conversion rate is built into the price of the goods/services
- conversion rate applies where USD isn't accepted, however prudent shopping of rates in both countries can minimize
Debit card
- less immediate risk because usually a PIN or signature is required for a crook to use
- tremendous mid-term risk because loss of the card can expose all the funds in your checking account to theft, possibly your other accounts if they're linked to the card as well
- low long-term risk since it's not common for fraud commited on your checking account to affect your credit report
- if debit card works at foreign ATMs, allows you to carry less cash at any one time, however increasing the # of ATM transactions increases the fraud risk
- debit card disputes can take slightly longer than credit card disputes, and not all banks provide provisional credit before the dispute is resolved, which means your money is gone unless and until the bank resolves the dispute in your favor
- may or may not be transaction fees and/or conversion rates depending on the bank(s), the merchant and the transaction
Credit card
- similar immediate risk to debit card, lower risk of PIN-based merchant or ATM fraudulent transaction, same risk of sig-based fraudulent transaction
- lower mid-term risk than debit card, because while you're arguing with the credit card company about your balance, you've still got beer money in your checking account
- higher long-term risk because of the potential damage to your credit report if the card is stolen, used and you cannot pay or get the credit card bank to resolve the dispute in your favor
- may or may not be transaction fees and/or conversion rates depending on the bank(s), the merchant and the transaction
There are other things like rewards programs and so on to consider. But hopefully this info, when added into the considerations about the expenses of the various payment methods, can help decide which makes the most sense for your personal tastes and needs.
One last item to note: not all atms and merchant terminals in the world use the US-style mag strip. In some countries, type 2 or other kinds of embedded-chip cards are becoming more prevalent. There are also countries (even 1st world countries like Japan) where cash is more the norm, and many smaller merchants don't accept any plastic at all. It always pays to do your homework well in advance of leaving.
Hope this helps.