If it is no, what alternative meaning do you give to 'spiritual' if it doesn't relate to spirits?
I use the word spiritual mainly because I cant think of any other word that accurately reflects what I'm trying to say. I lament the propensity among atheists to view the world mechanistically (I'm thinking mainly of how people mistake Dawkins' academic approach to how he is in person). When I am out walking in the woods I frequently experience feelings that must be akin to what religious people feel. Maslow refered to them as "peak experiences" and they are not confined to the religious. For me it is the stripping away of superstition and illusion that reveals the wonder around me. I sound like a tree hugger now!
You are not alone in this at all. One could of course make up a word for this but alas language is not so easy and people will just look at you funny if you come out with a new never heard before word. Currently, like it or not, "spiritual" is one of the best words available for this range of discourse and is the one most likely to engage the listener and let them know what you mean.
So I too find myself forced to use the word but as pointed out I do not mean spirits, souls or any of that when I use it. I use it to describe merely the experience of being human, of our shared human condition with other humans, and of our feelings related to our "place" in the world... and the feelings within us that the world is capable of evoking.
The feelings of interconnectedness and profundity that we attain fleetingly in our lives are interesting, useful and powerful experiences and they are worth exploring. That exploration is what I would be talking about when I talk of "spirituality".
There's some interesting work being done by people like VS Ramachandran on the area too. He has identified a part of the brain... by studying people in whom that part was "broken" such as sufferers of Capgras Syndrome... which control how relevant to us items, places and people are in our mind. By hyper stimulating this area he has illicted in subjects a feeling of everything being connected, everyone and everything being important and powerfully relevant and more. The way they describe the feelings afterwards are almost indistinguishable from the contemplatives who come out of caves espousing much the same notions.
As distasteful as it is that we are compelled to stick with the word "spiritual" I think it an error on behalf of some people in our world who abandon all interest in such realms of discourse solely because of their distaste for the label we are stuck with. As Sam Harris would put it, there is a range of attainable human experience out there which most of us are not even aware of and even fewer explore and even fewer again actually reach. There is likely some benefit in exploring that realm however.
Alas many of the things people say when having such experiences can often be hard to distinguish from magic and woo. The subjects of Ramachandran had the feeling everything was connected, everything was relevant, and it was all the expression of one supreme interconnected entity. That of course sounds like "god talk" but it is not really and it is worth exploring.
Of course it is only when reaching the end of this post I realise how horrible off topic it all is