I have only heard the LCD-3 very briefly so couldn't offer any impressions there, but I do own a pair of T1 and did some A-Bing with MHP1000 so I'll share my findings. As always, purely subjective impressions - that should go without saying, but folk sometimes seem to take things personally so I add the caveat! I'm also aware that several factors influence impressions, from choice of music to choice of source gear... it really is very subjective.
Anyway, the first thing I noticed going head to head was that the 1000 had the edge in terms of transparency - the T1 sounded a touch grainy by comparison. Also, with certain recordings, I find the T1 can occasionally sound a little too bright. The 1000 by contrast, always sounded smooth at the top end, but never lacking in detail.
The T1s treble extension is greater - at least according to the specs 5Hz-50kHz for the T1 compared with 5Hz-20kHz for the 1000 - but since the limit of human hearing is 20kHz, extension beyond that is surely irrelevant. That said, I suppose the real question is where roll-off occurs with the respective 'phones. I'm not aware of any FR graphs for the 1000, so all I can say is that I didn't notice any treble roll-off - the high frequency reproduction was incredibly detailed. In terms of bass extension, I'd say they were pretty similar, both going extremely low, but I felt there was more impact and body with the 1000.
Overall, the 1000 just had a tighter, more refined yet impactful sound but not really at the expense of having a narrow stereo image as can often be the case with closed backs. I would definitely say I perceived the T1 to have a wider, more open sound (hardly surprising) and that is perhaps its trump card, but the 1000 is still surprisingly open-sounding.