Alright, I haven't done enough listening to give an ultra in-depth review, but I'm gonna try to sit down and knock down some broadsides.
Soundstage:
The LCD-3's soundstage is larger than the DX's. Neither can has the largest soundstage around, so it's a little like fighting over 4th place, but the difference is noticeable (more so on some tracks than others). I'm not really sure how to describe the difference in soundstage, but imagine holding your arms out to your side so you look like a capital "T". With the DX1000's, the sound would be coming somewhere between you tricep and elbow, whereas the LCD's would be about mid-forearm to fist (I hope I didn't just confuse anyone with my overly technical description). The tendency for the DX to have sound closer to your ear can make some music sound more engaging, while making others lack subtlety. On the other hand, the LCD's can give some songs much needed depth, while making others feel a little further away than they should be, but I'd give the LCD the nod.
Sound Signature:
I'd say I'm fairly sensitive to high frequencies and shy away from bright cans, but I've yet to experience any noise fatigue with either of these. The LCD's are dark and smooth, while the DX's are closer to neutral (whith a slight lean towards dark) and have a crisper texture to the sound; not in a way that makes them bright, but you're more likely to hear fizz and pops with the DX's than the LCD's. I guess the difference could be described as laid back vs lively. The DX's seem better suited for energetic music, and the LCD's for more relaxed fair.
Bass:
The bass on these two are very different animals. The DX is more mid-bass focused and has a noticeably harder punch, while the LCD extends much lower, has more detail, and is more focused. Now, I say "focused" because the DX's bass tends to bleed over into other frequencies slightly. With the LCD, the bass is a thing onto itself, keeping other sounds seperate, but the DX infuses its bass into some of the song slightly; nothing bothersome or extreme, but enough to mention. And while the LCD definitely has punch, the difference between the two is like the difference between a really...... really hard push, and an actual punch. It basically comes down to which youd rather have: sub-bass detail, or slam? Fun or accuracy? They're both good in both departments, but have a trade off in those areas.
Overall, the LCD is technically superior and has a more even sound. Example being In the Hall of the Mountain King. The song speeds up towards the end, during which string instruments play rapid notes. Each note is quick and wavers just a bit, which was easily noticeable on the LCD's. This is a nitpick, however, since much of their detail is the same. It's only after swapping back and forth rapidly that some of those minute differences emerge. For most music (depening on your tastes), they'll give about the same amount of detail. The biggest difference between them is how they present the music as opposed to the amount of music that they present.
My personal pick for sound quality is the LCD, the main factor being it's ability to dig deep and give detailed sub-bass. Also, I feel that it's darker tone and distancing gives it a richer sound. That, and they are more nuanced, giving those small, extra details when available). With that said, I love the impact of the DX's bass and their comfort is a big win since they're much lighter and don't squeeze my head like a vice. Plus, they're much easier to drive. This might not be a big deal for some people, but the DX1000's give me great sound without makeing me feel tethered to an amp, which is a bonus. So other than the occasional album I've really been looking forward to, I'd feel comfortable saying I'd choose the DX1000's for most of my listening.
If you really like live/instrumental music (opera, orchestra, jazz, blues, etc) then the LCD's would get my vote. Otherwise, I'd go for the DX's (pop, hip hop, electronic dance music, anything with artificial bass). Rock, however, was a split for me... sort of. Stevie Ray Vaughn.... LCD. Def Leppard.... DX. Basically, studio vs stage.
Anyways, hope this helps. Let me know if you have any questions. I'm sure there's much to clarify.