I have a bit of a different story than a lot of people. I first heard DT in the mid to late 90s and wasn't all that impressed and don't even remember what songs I heard. Later, I felt like getting out of town and some friends were driving to Milwaukee to see DT so I went along. In the lead up to the show, I heard some of Scenes from a Memory. I didn't love it but I was intrigued and kept going back to it. My favorite music is usually composed in a minor key so when I heard Overture 1928 I was surprised I didn't hate it. I spent the next year listening to Scenes from a Memory off and on but I never really "got it" although I did appreciate it and liked a lot of the melodies and strange time signatures.
Over the next few years I scoured the used CD stores and got the rest of their catalog. My reaction to the rest was about the same. Definitely not my favorite band but I sure did like it. Then when I had it playing the background, Anna Lee came on. I had heard it before and usually skipped it but this time it finally spoke to me. I fell in love with it. Later I listened to Hollow Years and had the same reaction.
So even though I definitely like DT's heavy stuff the most now, it was actually some of their softer stuff that turned me on to them. Train of Thought came out and I finally "got it" and by the time 8V came out, I was a hardcore fan.
In summary: I've listened to DT for the better part of 15 years but for some reason only became a hardcore fan for the last 10+