Permissions are defined by the operating system. If you bypass the operating system, permissions are irrelevant.
You can't have access control (permissions) without authentication (determining whether the person requesting access is authorized). For example, to enforce that a file is accessible to root only, you need to determine whether the user is root. This determination is only meaningful within the context of the original operating system.
To have access control that's inherent to a storage media, you need a form of authentication that is external to the media. To protect data on a disk outside the context of the system it's connected to, your access control cannot rely on any information that's on the disk. For example, you can physically restrict who has access to the disk. If you assume that this form of access control can be violated (a stolen disk), you need something else. Access control via cryptography relies on two things: mathematics, which is immanent and cannot be bypassed¹, and the knowledge of some secret (password or key). So you can have access control using cryptography that doesn't depend on the media being used in a particular system, and in particular keeps working if the disk is stolen.
To keep files confidential in this scenario, encrypt them. An attacker with a stolen disk can make brute force attacks limited only by how many processors he devotes to the task (the disk speed isn't a limited factor, because the thief can make as many copies as he likes of the encrypted data). Therefore, be sure to use a strong password that the attacker won't easily find by repeated (automated) guesses. With proper software, you can reduce the number of guesses to a handful per second per CPU.
If you give access to the Linux system from a Windows system via a network protocol, it's a different story. What access will be granted depends on the settings of all the systems involved: the system containing the disk, the system where the user is, and the system containing the authentication data (e.g. a domain controller) if that's a different system.
¹ Barring major mathematical discoveries, but you can safely assume that won't happen. Cryptography (if done right, which is hard) is more robust in this respect than, say, armed guards (who can be bribed or recruited).