If the former tenants are still living there when you tour a rental unit, observe how clean (or unclean) they are. A dirty kitchen will, of course, attract both rodents and bugs. But if you see papers everywhere, or spot holes in their couch cushions and carpet, they've also given rodents plenty of fodder for nesting material.
Rats and mice love to make nests out of whatever they can find. Their favorite materials are paper, cushion stuffing, tissue, hair and cloth. Take a flashlight with you when you visit the rental unit and check all the nooks and crannies for red flags like shredded paper. Don't forget to check inside cabinets and behind appliances in the kitchen.