1. A computer-implemented method for detecting suspicious attempts to access data based on organizational relationships, at least a portion of the method being performed by a computing device comprising at least one processor, the method comprising: detecting an attempt by a computing device within an organization to access an additional computing device within the organization; identifying, based on a directory service associated with the organization that classifies the computing device and the additional computing device, an organizational relationship between the computing device and the additional computing device; determining, based on the organizational relationship between the computing device and the additional computing device, that the attempt by the computing device to access the additional computing device is suspicious due at least in part to a device priority level assigned to the additional computing device exceeding a threshold device priority level; performing a security action in response to determining that the attempt by the computing device to access the additional computing device is suspicious.