I really like Tchaikovsky's piano music, although it had been explained to me that it is a bit awkward to play since he himself was not much of a pianist, and indeed there is none of the natural smoothness of Chopin in how the hands move when I play his music. Also, I find his pieces really hard to memorize (at least that goes for the Seasons which I tried to play recently, I wanted to learn one piece every month in 2014 but stopped halfway, on the June barcarolle... I'll try again next year). My final complaint is that I have yet to find recordings of some of his music (like the Children's Album) that I like, I think much of what is out there is played too fast and without much love for the material (although I have a recording of The Seasons by Vassily Primakov that is simply superb).
But there's something "Russian" in his music that appeals to my Russian soul. Even though he never positioned himself as a specifically Russian composer, his music seems to get much closer to my heart than his German contemporaries like Brahms and Schumann, perhaps he uses some harmonic transitions that are more common in Russian folk music that is deeply ingrained in me... I guess I'm not much of a music theorist, so I'm just stating the fact - playing Tchaikovsky is for me an act of going back to my Russian roots, and as I work on his pieces I often discover depths that I fail to hear when someone else plays them.
Here's a video where I play several pieces from the Children's Album:
Why these specific pieces? I don't really know, this is what I learned a few years back and now play as warm-up practice... but I do think these are very enjoyable musically. In particular, I think all his "foreign songs" are very good reflections of how Russians view music of other peoples.
I have big plans for Tchaikovsky but not for the immediate future. Next year I hope to record something from the Seasons and perhaps other pieces from the same book (I got a very good collection of his piano pieces).