I think you guys have hit the nail on the head:  Many evangelicals seem to have lost confidence that there's really anything of value to be learned from the Bible. 

As for how to fix the problem, let me add just a couple of things to Mike's post below.  One is that if you want to teach your people that the Bible is more than just a compendium of pithy sayings--as if the whole thing was meant to function like Proverbs--you have to preach longer sections of text than just one verse.  And you certainly have to do something other than skipping around the Bible from week to week.  Part of teaching your people about the beauty and power of God's Word is to show them its shape and structure, to tell them how a book flows, what its emphases are, and how it functions as a whole.

I preached a sermon series through Isaiah some time ago.  Before I started, I'd never really understood before how Isaiah fit together, or where it was going.  To me, the book always seemed absolutely enormous and entirely unwieldy.  Sure there were parts of it that I'd camped on before--birth to a virgin, death of the Servant, etc.--but that meant I gave short shrift to the parts didn't consider so important. 

So I decided this time that I'd approach the book differently.  I treated the book like a sword---picked it up, felt its weight, learned how it balances and how each of its parts contributes to the whole, realized how through all those scores of chapters the whole thing is really coming down to one razor sharp point.  In short (and perhaps to push a metaphor too far), I learned how to wield the book (I hope) as the Holy Spirit meant it to be wielded when He forged it.  My goal then in preaching was to hand that sword to my people and teach them how to wield it, too, to feel its weight, trace and understand its contours, and feel its balance.  I didn't want just to stare myopically at the hilt, or just point out an inch or two of the blade and then skip back to the handle.  I wanted them to know and become familiar with the whole sword, and equip them to use it in their lives.

That's just one thought on this whole topic.  Whatever you do, and however you preach, your goal needs to be to show your people the beauty and power of God's Word.  And you're not going to be able to do that if you act like you're embarrassed by it, or if your service communicates in whatever way that Scripture is boring.  So read Scripture in your services---more than one or two verses, and without music.  Preach more than one verse at a time.  Don't preach sermons where your people can read along with you at the beginning and then close their Bibles.  Preach the kind of sermon where they constantly have to look back at the text.   After all, we want our people to know the Bible's power, right?, not just confess it.