Indeed, some of the most spiritual people do not follow any of the larger religions.
Native Americans, medicine-people and shamans, for instance are heavily spiritual. In fact that is usually the entire basis of what they do. The communicate with the spirit world, seek knowledge from them and bring it back to their peoples.
Spirituality does not necessarily mean religion in any sense. It simply describes a deeper connection to a non physical world such as ours. A connection to something greater than the physical plane we are on in which the concerns primarily centre around material things. A spiritual person is less concerned with physical things in general and more concerned with the metaphysical.
It is often tied to a religious belief by the concept of spiritual meaning soul.