Hmm... this might seem like a strange recommendation but:
The Legend of the Mystical Ninja (Capcom)
http://www.youtube.com/watch?v=3KLjIXsMIJ8#t=01m38s
The first
Gambare Goemon game to be localized for western audiences. The game is split into 2 modes: isometric exploration where you search for shops, beat up random hoods for cash and gain power-ups; and platformer-type stages (in lieu of dungeons) where you traverse a linear, side-scrolling path that leads to a boss you must defeat. Top-notch graphics for the era and an excellent soundtrack. Can get right difficult at points. Has 2 player co-op and a password system to continue your progress (no save batter/SRAM).
Not exactly Zelda-like except for the overhead map portions (unless we're talking Zelda II, and then maybe only a little), but it's a very fun game and chances are you'll enjoy it as you said you like fast-paced action.
Also straight-up echoing
Soul Blazer recommendations, that is an excellent game that predates Illusion of Gaia/Time, and from the same team.
Final Fantasy Adventure/Seiken Densetsu is also a great Game Boy title that will be right up your alley.
Furthermore, since you mentioned PC might I suggest giving the
Ys games available on Steam a go? They skew a bit more to the jRPG side but you won't find any turn-based battles here, just pure unadulterated quick-reflex game-play.
Ys I&II Chronicles+ feature the old-school bump system, but
Oath in Felghana and
Ys Origins has more traditional game-play systems. Do it for the soundtracks alone, they are bliss.
On the Playstation side there's
Brave Fencer Musashi as mentioned above,
Blood Omen: Legacy of Kain if you like your adventure games to skew towards the Gothic (and hell, why not
Soul Reaver while you're at it?), and perhaps even the
Mega Man Legends series... though they lean more towards PEW PEW than what you would expect from a Zelda-like.
Lastly, though full-blown RPGs with turn-based battles and what-not, the
Wild Arms series (especially the first three) features puzzle-solving in most of the games' dungeons using a unique set of tools at the disposal of each of your party members.