A few from different platforms. This list doesn't necessarily represent my favorite games, but represents the ones I think have the best music. Each links to one of my favorite songs from each game on the off-chance you've never heard of them.
SNES
F-Zero- Better than Mario Kart in all respects, not just musically.
Yoshi's Island - Remember that Mario game where you piloted a submarine that fired heat-seeking torpedoes and the final boss had a metal soundtrack? Plus it had that sinister waltz in it and I love waltz music. Hard to choose, whole soundtrack is awesome!
Donkey Kong Country 2 - For all intents and purposes, musically this was better than the first game.
Stunt Race FX - A seemingly juvenile game at first, but surprisingly fun and with a good soundtrack.
Uniracers - Who said the SNES can't do guitars? Hah, jokes on you.
- I was tempted to put Killer Instinct on this list along with some others like Street Fighter II (World Warrior, the other versions suck). The SNES is my childhood system and by far my favorite.
Mega Drive
Outrun 2019 - Impressive, even has drum solos. Takes some balls to do that with a Sega.
Streets of Rage - Loud and obnoxious, the intro to a Dreamland song (Anything for you) always reminds me of it too for some reason - about 38 seconds in.
Road Rash III - Rule Britannia!
- I am not a huge fan of the scratchy racket the SMD makes, I was but I started to dislike it more and more when I had to work with the hardware for extended periods of time. The Yamaha DX series were just shit. So I have a lot of respect for people who could get good sounds out of this thing, and a strong dislike of people too lazy to use the sampler or the PSG like the idiot who did the Thunder Force 4 soundtrack. What a loser.
Also, sorry, but I like the Michael Jackson game's music too. Actually, the game was kinda fun. Oddly, I don't really like Michael Jackson's music that much.
5th Gen
- Mostly crap, but it had a few good ones. So I have them all under one heading.
F-Zero X - and the Guitar Arrange versions of the songs Nintendo released on CD. Hey kids, this is the face of Nintendo I grew up with. (N64)
4x4 World Trophy - Because who wouldn't want to listen to obnoxious rock music whilst driving trucks? I like it anyway. I've owned some of the albums from the game since before I owned the game. (PS1)
Sonic 3D Blast - What a mediocre game (Seems to play better than the SMD version) but the OST was good in places. Very 90s. (Saturn)
PC
Duke Nukem II - I'm a Megadeth fan, so this goes without saying. Linked to my own recordings.
Atlantis: The Lost Tales - This was just magic, the theme is also good.
Atlantis II - OK, why does nobody ever mention these games? Again, has more good music in it than just that one.
Terminal Velocity - If that's no the 90s, I don't know what is.
Commander Keen - Because Bobby Prince is awesome anyway, part of the reason I started playing with synths.
There are more, but this is just a short list.