Must Be Played Old Open World Games

Gothic 1&2, the 3rd is meh. But the first 2 have a special place in my heart.

Brutal Legend is awesome and has a 90% discount right now !
Don't be fooled by Steam, the game was launched in 2009.

Overlord is great too and has 75% discount.
Check the DLC too.

FF12...

Best I can suggest is San Andreas (Im assuming you already checked it out since you mentioned GTA3), Morrowind, Fallout NV, and Fallout 3. Fallout 3 needs a bit of tweaking to get working on newer systems, but its very, very worth it.

Overlord 1 and 2
Delta Force games
Tom Clancy's Ghost Recon games
Fallout 1, 2, Tactics, 3 and New Vegas , and the upcoming Frontier mod for NV
Arcanum
Saints Row 2 (not 1 since it's console exclusive), 3, 4 and Gat from Hell
The Sabouter
The Godfather 1 and 2
Mafia: City of Lost Heaven
Driver: Parallel Lines and Driver San Francisco
Crysis 1 +Warhead expansion
True Crime: New York
All GTAs
Red Faction Guerrilla Steam Edition
The Witcher 1 and 2
Far Cry franchise
Assassin's Creed games
I am Alive (okay, not really or fully open world, but you get to traverse an amazing but tragic looking world on a set path, but feels like a world of sort but if you want pure OPEN world then skip this :( but I'm recommending it for the atmosphere and mood )
The Lost World: Jurrasic Park
Bunch of good old MMOs that are alive?
Just Caus 1 and 2
Dead Island Definitive Edition and Dead Island Riptide Device Edition
Dead Rising 1, 2 and Off The Record
Scarface: The World Is Yours
Defiance
The Warriors (semi open world)
Prototype 1 and 2
Burnout Paradise City
Midnight Club franchise
Bunch of Tony Hawk Pro Skater games like HD I think, Underground, American Wasteland and maybe more, don't remember...
I wish Mercenaries was on PC, that seemed open world and fantastic, but that's all I remember...

mercenaries for ps2

Prototype, best purchase during the sales!

It's not technically open world, but Hidden and Dangerous 2 (currently on sale) has large open levels that let you decide what approach to take to get the mission done rather than running down a corridor with scripted events like so many games now. It uses the same game engine as Mafia I, as well as driveable vehicles.

Postal 2

These do not fit into your timerange... but it is really close to it.

• Legacy of Kain: Soul Reaver (1999)
• Darksiders (2010)

A others mentioned Total Overdose and The Gun are quite similar to GTA3.

Outcast. A bit older than 2000, but very playable. I really liked the main character and had lots of fun with it. There's no hand-holding about where you need to go for side quests and stuff; you just have to talk to everybody.

The Saboteur. It's one of the best video games ever.

Old open world game? Betrayal at Krondor of course!

all Saint Row series, starting form SR2 wich have the best story ..there is Sleeping Dogs wich is a hidden gem too.those are open wirld games..fallout 3 -new vegas.skyrim too.prototype is a fun game too.. as batman series.

ArmA: Cold War Assault (formerly Operation Flashpoint Cold War Crisis) - hands down the game you are looking for! Released in 2001, and one of the first (or even THE first of all) real open world games where you were able to pick up anything your enemies dropped, able to use all vehicles .... god dammit it was the greatest sensation of my entire gaming life! =)