The first thing to consider is that "Native American Spirituality" as a term implies a cultural uniformity of the First Nations that simply didn't exist. This tendency to think of a single "Native American" culture is a result of racist thought, wherein things like skin color and hair texture are taken to be the sole determinants for identifying someone's culture, ignoring concepts of tribe or nationality. The reality is that a Pueblo, a Hopi and an Aztec can each be called "Native American," but their religions aren't likely to look a thing alike. Similarly, a Scot, a Basque, and a Slav can all be thought of as "White" or "European," but their cultures and religions are a diverse set of animals as well.

The second thing to consider is that these spiritualities tend to be community based rather than conversion based. This means that to legitimately be a member, you must be a member of the community in which the religion dwells. This is something of a foreign concept to Christians, but should ring some bells with Jews.

The third thing to consider is that the people who are legitimately members of these religions don't necessarily want you. There's a history between European cultures and the Native American Nations that can't exactly be described as one of trust and friendship, and that unavoidably informs and defines the context in which Europeans are going to be regarded by those they approach, especially if they do so with an attitude of ownership, entitlement, and co-optation.

But enough introduction on my part:

Myke Johnson had written a much more extensive and detailed discussion of this topic, with an eye also to what to do if after you've understood the issues entailed you still wish to proceed along this path, as well as an exmaination of some of the ramifications of having started along this path without that community sanction. It's moved since I first linked to it, so I've removed the broken link, but the site itself is worth looking over as well.

The Lakota are fairly direct about their stance on the whole thing. They've formally declared war on those who appropriate their ways rather than enter into their Mysteries legitimately, if I can apply some Western terminology.