Without a wide variety of Sega games, gaming's descent into COD circlejerks and "WOW CINEMATIC" gaming was probably sped up. Sega didn't lose it all with the end of the Dreamcast though, and their classic style did continue for a time on the PlayStation 2, Xbox and Dreamcast. This is a list of the games that snuck out of Dreamcast development (or were made shortly after the end of the Dreamcast and before the Sammy merger) and had a second life on other consoles.
I'm not going to include games like Shenmue II or Rez since those did get a Dreamcast release (even if it wasn't necessarily in the US). I'll also be excluding sports games, such as the excellent NFL 2K series.
Super Monkey Ball
An early Gamecube game, Super Monkey Ball is actually an upgraded version of an arcade game. The arcade version was known as Monkey Ball and was only released in Japan on Sega's NAOMI. The NAOMI was basically a Dreamcast and it even featured a GD-ROM drive. The NAOMI's major difference was that it only read the GD-ROM at boot; it loaded the entire game into RAM to eliminate load times. Many Dreamcast games which have an arcade counterpart ran on the NAOMI board, which is why they're such perfect ports. On that topic, a pet peeve of mine; Soul Calibur's arcade version DID NOT use the NAOMI. It actually used hardware which was based on the PSX, and it looks like asshole.
Virtua Fighter 4
There's actually not a whole lot of information out there on Virtua Fighter 4 and any Dreamcast incarnations. The Japanese limited edition of Shenmue II included two Virtua Fighter discs, one with a history of the series and the other with some videos of Virtua Fighter 4. It's unclear if any work was ever actually done on Virtua Fighter 4 for Dreamcast. The game used the NAOMI II hardware, which was beefier in the RAM department and probably would've had a rough time on the Dreamcast. My gut instinct says that it must have been planned for Dreamcast at some point, or else including these bonus discs in Shenmue II was just downright cruel.
Jet Set Radio Future
Jet Set Radio Future is an excellent game where you play as members of a rollerblade gang (wut) and have to tag walls. JSRF was an early Xbox release, but didn't help the console gain much traction in Japan. The game oozes that classic Sega je ne sais quois and has an excellent soundtrack to boot. It's similar to the first game, but adds a multiplayer mode which is pretty fun. Unfortunately, there's no LAN option so you can't take it onto xlink kai or xbconnect. If you've ever been curious about the series, the first game recently got re-released on every platform known to man. JSRF is a more fluid game, but the first is worth revisiting (or visiting) as well.
Sega GT 2002
The first Sega GT on Dreamcast was basically a shameless Gran Turismo clone, but it was a lot of fun. It was actually my first Dreamcast game, and I didn't have a memory card. So I left it on for a week so I wouldn't lose my progress. Try doing that with an Xbox. While the first game was excellent, I barely played this one. I remember the control being pretty stiff and the cars not handling well. I will say it has been a while since I played the original, though.
Another game that I haven't played. Anime games are generally shitty affairs (as anime often is), but the Sega logo is enough to make me curious since at this point in time it still indicated some level of quality. Maybe some day I'll check it out, the reviews seem decent enough.
Panzer Dragoon Orta
In my opinion, the Xbox got the best bunch of Sega games. Panzer Dragoon Orta is a great example of this. It's a Starfox-style game which uses a made-up language (also Starfox-style). It's not a particularly challenging game, but it's entertaining and uses the Xbox hardware well. The game also includes a port of the PC version of the original Sega Saturn Panzer Dragoon, which is a load of fun since it's somewhat difficult to justify an investment in a Saturn (in my opinion). While there's no word on a Dreamcast version of this game, I think it's a given. I can't imagine that Sega would have started development on the game and thought it would find a huge audience on the Xbox, especially with its flop in Japan.
I'm actually playing through this game right now and it is awesome. The soundtrack is excellent and the control works well (even if the camera is a bit iffy). This is a game that was in development at some point for the Dreamcast, and it sort of shows; it's not necessarily a graphical powerhouse. The Dreamcast and PS2 were actually pretty evenly matched due to some design oversights by Sony, RAM aside. While later PS2 games would impress (for the hardware), this game seems to have been borne of weaker origins. I really can't imagine playing this without a second analog stick, though.
The first time I saw this game was actually on an X-Play review (I believe it was still Extended Play then), and I'm pretty sure Sessler just bitched about how hard it was the whole time.
Another game that was confirmed to have origins on the Dreamcast (there's a few poor quality screenshots out there), and another game that I haven't actually played. I did own a copy of it, but I never got to play it. In dumber times, I left my Xbox with an ex-girlfriend, and trusted that Microsoft would emulate the original Xbox well enough on the 360. This game never made it over, and I ended up selling my copy when that girlfriend became an ex. I actually did get that Xbox back eventually, but never picked up another copy of Gunvalkyrie. The Dreamcast version was said to use a control scheme that combined controller and light gun usage, which would have been interesting.
The House of the Dead III
The House of the Dead III continues the classic arcade shooter series and gives you a shotgun this outing. The game isn't as immediately addictive as The House of the Dead II, but it is still an excellent title. I played through it several times, and you can unlock The House of the Dead II as well. There's not a whole lot to say about it due to the nature of the game, but the Dreamcast version was planned to be cel shaded. It would've been an odd shift for the series, but likely one that would've went well.
Crazy Taxi III: High Roller
Crazy Taxi is a game that everyone has played at some point, and it has one of the most memorable soundtracks in game history (YA YA YA YA YA). While the first game is incredibly addicting, I don't think the second or third entries were as fun. The additions to the gameplay (jumping) took the simple fun away. I don't think this is an uncommon feeling since Crazy Taxi is available on every console known to man, while the sequels remain trapped on their original consoles. I used to have this game, and I can't say I played it too many times.
ToeJam & Earl III: Mission to Earth
A sequel to the classic Genesis games... and another that I haven't played. This game is confirmed to have been in development for Dreamcast, with a beta recently being leaked which is playable on consoles. It received mixed reviews, but it's another game that I certainly intend to pick up at some point.
I'm leaving off some games (Billy Hatcher and the Giant Egg most notably) because I either haven't played them, or they're compilations/remakes. Others are being left off due to coming out after or near the Sega-Sammy merger. The merger was a turning point for Sega, with Sammy largely gutting the company to focus on pachinko machines (yay). Either way, if you never had a Dreamcast or don't want to, these games should give you a taste of what Sega used to be like. In addition, a large amount of Sega output during this time period was ports of Dreamcast games to other consoles.
The Sega-Sammy merger? That's why Sonic kisses humans now.
Look at Knuckles back there, he's about to stroke his echidna peen.