Yes. People play UT 2004 online.
However--they play the two game types that UT 2004 was best for -- Onslaught (excellent, great fun) and Invasion. The player counts are not huge, but you can still find packed servers at most times of the day, especially during North American and European prime time. Onslaught is the vehicle game which is somewhat similar to DOMination in principle, where you try to take over control points. The vehicles are a lot of fun and this is the game that made UT 2004 IMHO. Invasion is a non-competitive shooting gallery game where the targets shoot back. The best Invasion servers are called Invasion Monster Mash RPG where the server remembers your player name and CD Key and keeps track of your player, so you can level your player up over time (more health, more damage ability, etc, lots of stuff) and also find magic weapons. It's a great game to play during off-prime time when other game types' have few players. The best Monster Mash is the Clan DW server. They also have a nice Onslaught server. The best Onslaught server IMHO is the Omnip)o(tents server, which packs up with 32 people during North American prime time. (Why do I keep advertising it?)
UT99 was best for DM, CTF, and DOMination. Domination is completely dead now. You can still find people playing UT99 CTF and DM but most of the populated CTF servers have mods such as all-sniper, instagib (one shot, one kill), low gravity, and Strangelove (ride a nuclear missile--which is fun). A lot of people still play UT99, actually. Experienced CTF players might be interested in the PUG--pickup games--matches which are like 5v5 clan matches but without set teams or scheduled times. They are organized on IRC (Internet Relay Chat) and people join voice channels for their teams. Those are played in regular weapons CTF and are a lot of fun.
UT3...it was a tremendous disappointment. UT 2004's Onslaught game is much more fun than UT3's Warfare game, and UT99's CTF and DM games better than UT3's CTF and DM games. Overall UT3 isn't too bad, but if you played the earlier UT games, you probably won't be too impressed with it. I'll play UT3 CTF now and then.
Why UT3 failed.
As with all UT's, legions of fans were looking forward to it. When the "Beta Demo" which is actually the final Demo, came out in October, the UT99 pro community liked the game play and even anticipated moving to UT3. However, it had some serious problems. The server browser was god-awful--it was featureless and barely functional and you couldn't even add Favorites! The User Interface was also god-awful. It felt clunky, slow, and consolized. Basically, it felt like the game was designed for consoles and then ported over to the PC.
Everyone figured that Epic would fix these things before the final release. Epic didn't and the Retail Release = Beta Demo. The end result? People got word of it and...didn't purchase it! People who did buy it soon lost interest. The game play wasn't all that amazing either. You couldn't throw your empty weapons (upon the game's release), for example and the user interface offered few graphics tweaking options. The server browser was also god-awful for the first few months until it was patched up. (As it stands now, it's not too bad but you still can't call it up easily while you're on a server to check for other servers, which took 2 seconds in the earlier UTs, instead you have to disconnect.) You also have to log in to Gamespy so if the Gamespy server is down, you can't play online. The game play itself wasn't all that great. It only had 4 game types--Warfare, Deathmatch, CTF, and vehicle CTF. The colors look washed out, as though it were Gears of War-ified and IMHO, the maps feel a little cluttered with all the eye candy. The CTF movement speed is also about twice as fast as in the original UT99, so you will often die instantly without having had any chance to react to anything, at least much more often than in the Original. They also kept the silly translocator throwing limit in.
Warfare wasn't as much fun as UT 2004's Onslaught and they removed first person view from vehicles, which is a big problem for flying vehicles and the vehicles don't handle well. They also made an awful, console-crowd driven decision to add an invisible vehicle that lays down a mine patch which you cannot destroy. (What were they thinking?) The addition of the Orb is OK and the hoverboard is an improvement. However, overall, UT 2004 is just more fun. The looks are better with the brighter color palette and the vehicle feel much better.
You can pretty much say the same thing for the other two game types. UT99 CTF is much more fun (and more populated overall) than UT3 CTF though UT3 CTF beats UT 2004's hitscan-based CTF. UT99 DM beats up on UT3's DM as well.
Epic's mistake was a failure to copy the feel and game play of the original UT99 exactly as it had been for on-foot-based game types. UT3 doesn't feel too bad really, but the default speed is just too fast. (All they needed was the UT99 movement speed but 10% faster.) They then needed to copy the feel of the UT 2004 game play for Onslaught and Invasion, especially the vehicles. They also needed to add some of the other game types, especially UT99 style Domination, Invasion, and Bombing Run, but that's not what really killed it.
I have also heard that it is harder to run a server for UT3, or at least people had many problems with it at the beginning and you couldn't have a Linux-based server either. Another problem is that producing custom content is much more difficult now. Whereas everything went into a single, coherent folder for UT99 and UT2004, instead they came up with a cockamamie system where maps get "cooked' and where you also have to pay attention to the obligatory Application Data folder. I tend to think that some of this is the influence of consolization. Regardless, the file structure is much more complicated and less-transparent than in the other games.
UT3 had an awful, horrible launch, and whoever made the decision to release it when it was released in the condition it was in should have been fired IMHO. Supposedly, they wanted to get it out the door in time to be on the shelves a couple weeks before Christmas. However, this was a very non-strategic time to release the game since other big title were coming out--Crysis and Team Fortress 2 (and maybe one or two others). That made it an even worse time to release a game that was essentially still in alpha or beta condition. The end result is that UT3 felt like an unfinished, buggy beta console port, so it never built up large player counts online nor much of a fanbase. UT3 was, in essence, a stillborn.
It's too bad because UT3 had potential. They did get a lot of things right with UT3. The overall movement and feel--perhaps the hardest thing to get right--isn't bad at all and the UT99 pro crowd liked it. (I still think the base movement speed is too fast.) I load it up and play a little CTF every couple weeks and when I do, I think to myself, "What a shame."
Did Epic learn anything from this? We'll have to wait and see what UT4 is like if there ever is a UT4. Last I heard, they were shelving the UT series for a while. If they ever release another UT, hopefully they'll go back to the winning formula they had in aspects of UT99 and UT 2004.