Aware but not overly concerned.
I don't recall ever having read on these forums [including many trip reports] of identity theft being a problem. It's not to say it doesn't happen, but I would consider it to be a rare and unlucky thing to happen to anyone. Certainly no riskier than it happening to you in your home town when doing the weekly shop or filling with gas.
You can buy 'Gift cards' for fuel that are basically a pre paid card you can use at the pumps. Most major chains use them. The only downside is you are tied to that brand and it would pay to get them for different brands in smaller amounts. With those, a bit of cash and paying for meals and lodgings on a credit card should see you pretty safe.