I could care less about puzzles in Zelda. They are a chore and a bore.
But I started with Zelda I and II as a kid. They had some basic push-block moments, but generally they were about running around figuring out what I'd call meta-puzzles -- taking clues given by NPCs and trying to figure out how to access new areas or secrets. No one told you where to go, at least not directly. The focus was exploration, not hitting switches. But the Zeldas after that just got puzzle crazy.
See, to me -- and I think Miyamoto's original inspiration of wandering through forests as a child -- the Zelda concept is about exploration and adventure. You're all alone on a quest. It's kind of dangerous and exciting and mysterious. And battling dangerous creatures -- for instance, Zelda II's combat system was great and wish they'd get back to tighter gameplay like that. In II, you felt like even random encounters were kind of a dangerous (but survivable) event.
So I think modern Zeldas kind of lost that spirit. Wind Waker had a bit of that on the high seas, but the controls were so frustrating that I really didn't want to go out of my way to explore. I get that kids who grew up on the SNES and Gamecube Zelda games think Zelda=puzzles, but I think it's a distraction from what made the OG games so fun.
The reveal of the Wii U footage has me tentatively optimistic that more action and exploration elements might be coming back to Zelda.
edit: I see some people above think the terms "adventure" and "exploration" are generic and don't mean anything, or just nostalgia trigger words. It's the mechanics, setting, and level of handholding which determine whether a game makes those attributes a focus or not. Modern Zelda games focus heavily on puzzles, heavily on making things too obvious. There's no "exploration" per se -- your journey is focused on taking you from one dungeon to the next one. Part of this is a bigger focus on narrative, which I have no problem with per se, but narrative creates structure, and exploration requires the opposite of structured gameplay.