The Pattering of Little Feet
In addition to searching for odd smells, you'll also want to keep an ear out for rodents. Because mice and rats love to hide behind walls or under floorboards, it can be difficult to spot them. But if you get the chance to listen attentively, you may be able to actually hear rodent activity.
Listen for sounds that seem like scurrying, scratching on walls or gnawing. It can help if you're able to visit the rental property at night, when the nocturnal rodents tend to be more active.
Aside from movement, you could possibly hear rodents squeaking. Both mice and rats squeak when they're fighting with each other or when they communicate.