Cyberpunk

The SNES Shadowrun game is good. I played it originally on SNES and few years ago on an emulator. All around good RPG. If you've played the new PC Shadowrun games from Harebrained, the lineage is clearly there. Systemshock is still a good game as well, but these days I would rather play the remake.

Neuromancer was excellent.

Not quite Cyberpunk, but if you like the Fallout vibe, check out the original Wasteland.

Lifeforce Tenka (or Codename Tenka) in NA territories was underrated.

A PS1 FPS that experimented with having just one gun, but could be modded to provide different weapon functionalities (or weapons in real terms), and the soundtrack was absolutely banging.

Classic Psygnosis really.

Speaking of whom, there's also Sentient - perhaps not necessarily good, but tried so many new and inventive things that are common in modern games, that it deserves it's own post really.

edit: in fact, how can we mention Psygnosis and not me tion G-Police? That was a fantastic game, with great graphics (let down by a short draw distance), awesome gameplay, and a absolutely amazing storyline. Well worth a play, especially for the particularly mental trailer.

Snatcher!

Edit: I just realized I didn't read the whole post, sorry. Snatcher isn't action-oriented at all. Well, except some short shooting sections.

There's a PS1 game called One which I think might be cyberpunk:

But unfortunately, either I'm terrible at the game or the game itself is terrible because I can't beat the first level.

I liked terminator for zx spectrum