I'm not sure a secular neuroscience necessarily leads to a spiritually impoverished reductionism with no interest in the deeper side of life. Some people worry that the scientific method reduces a rich inner life to "just another brain state" but among neuroscientists there's usually considerable awe at the enormous complexity of those brain states, and how very little we know about them. There is also an interest in deep inner states such as meditation - see this video for the interesting history of that side of neuroscience http://www.bu.edu/phpbin/buniverse/videos/view/?id=247
I think a secular world-view will however change our understanding of ourselves and our inner lives profoundly; much much MUCH more than secularists realize or are willing to admit. Thoughts, feelings, decision-making, values, personality traits, morality, existential judgements, love and fear will no longer seem to inhabit a distinctly mental realm, but will be described neuroscientifically in immense detail. I don't think the terms will go away: we all have a 'theory of mind'; we can all take the 'intentional stance'. But as neuroscience progresses people will come to accept that they are brains; that their inner states are brain states, no more no less. My opinion is that this will lead to (1) widespread moral relativism, not in the sense of anarchy or cruelty but if moral values are well-defined, vividly visualized brain states explained in high-school biology textbooks then nothing IS right or wrong, it just SEEMS right or wrong; (2) widespread adoption of techniques to change the brain, driven by and driving neuroscientific advances and moral relativism; and (3) a situation where a scientific understanding of people in general and yourself in particular will be extremely valuable, not displacing subjective forms of knowledge but rather enhancing and being enhanced by them.
Where does this leave spiritual experience and the search for deeper meaning beyond the material? I don't know. Maybe, as traditional organized religion gradually acquires the status of myth and make-belief not suitable for a scientifically aware 21st century person, civilization will see an even greater hunger for transcendence and religious conviction. Maybe not. Maybe you can fill that hunger with ever larger TV screens and new technology, maybe not. Maybe we will turn our eyes toward a medical and technological defeat of the aging process and explore the stars together.
Part 2, part 3