I've heard that a lot too, and from intelligent people, mainly Unitarian Universalists. It seems they understand religion as an authority-sanctioned creed or dogma, and they do not want to do anything with it. But they do not want to give up their belief that there is something larger than the individual or even sum of individuals, and that something larger is interconnectedness of all beings, humans, animals, nature.
This belief is really a form of pantheism - perhaps the most hated "heresy" by mainstream religions - whose main tenet is that there is no distinction between the sacred and the natural, or perhaps that the nature as a whole is a form of the sacred. However, the concept of "sacred" or even "-theism" in the term pantheism sounds to Christian to these folk, so they seem to prefer "spirituality" - which definitely has Wiccan, pagan and non-christian connotations. Conventional religions (especially Catholics) hate pantheism because it undermines the raison d'etre of the priestly class, who claim to be intermediaries between the natural and the sacred. If there is no difference between the natural and the sacred, who needs an intermediary?
I have to admit that I view pantheism i.e. its central tenet of one world that is both "natural" and "spiritual" as quite intellectually appealing, more so than positivistic scientism. Positivistic scientism is, after all, a dogma that shuts the door for certain forms on inquiry (namely those not accepted in current scientific practice), whereas pantheism keeps all the options open, even those beyond the current level of scientific knowledge, without falling into the trap of religious dogmatism and fundamentalism.
Another thought - getting rid of the concept of creed or dogma seems to be an interesting response to the challenge to religious authority posed by a democracy. Religious authority is needed to maintain coherence of the religious system of beliefs - logic alone is insufficient to answer all inconsistencies and contradictions (cf. the Goedel theorem) - and executive decisions must be made to settle ambiguous issues. In the Catholic and spun-off churches that executive power resided with the chief officer of the church e.g. the pope or the King of England, who was deemed "infallible" - which simply means that his decisions in the matter of doctrine are final and not subject to any further challenges, pretty much as the rulings of the US Supreme Court).
However, the supreme human authority is incompatible with people who internalized the principle of equality or democratic governance - such as the US colonists, or Muslims. Yet, giving up the notion of supreme human authority comes with a price - opening the door for uncertainly and ambiguity resulting from insufficiency of logic to settle doctrinal inconsistencies and contradictions (that pesky Goedel theorem again). Som,e people, like Unitarian Universalists, had no problem with such a situation and made the next logical step of jettisoning the notion of creed and doctrine altogether.
However, many people have difficulty dealing with ambiguity and uncertainty (which I believe is a form of mental disorder) - they need certitude. So how do you restore certitude without a recourse to a supreme human authority? By insisting on literal interpretation of the religious scripture aka fundamentalism. This may explain why fundamentalism has considerable following in populist societies (such as US or Muslim), but has little traction in Europe which retained the conventional notion of religious authority vested in the chief officer of the church.
In sum, pantheism or "spiritualism" (which are really two words referring to one thing) is the renunciation of the conventional religiosity based on dogma, authority, and certitude without falling into fundamentalism or positivistic reductionism.
PS. Perhaps the most famous European pantheist was Baruch Spinoza, a 17th century Dutch Jew (no coincidence here, since kabalistic Judaism is a form of pantheism as well).
Wojtek