Thanks, @sam9s and @vaibhavyagnik!
The problem has been solved.
Now don't flame me, the culprit was the power supply. I don't know why my regular office PC's PSU did not work when I used it in the new PC. Probably because the old PSU has a 4-pin CPU-power connector and the new motherboard needs an 8-Pin CPU-power connector, and probably because I fitted the 4-pin connector in a wrong way or something. Clumsy me
Today evening, I got (= armtwisted) a friend to come over with the CPU of his newish machine, based on the Core i3 2100 processor (same as mine). I installed his processor on my board and the problem persisted -- no startup. Needless to say, I was
very frustrated. Since I had a fully working donor PC in my hands (I must thank my friend for letting me butcher his new build), I started swapping out stuff one by one, onto my PC. First I tried the RAM, and then I tried the PSU. My PC started right after the PSU was swapped. Problem solved.
I've got in touch with theitdepot about warranty support on the Corsair PSU. They will probably call me tomorrow with the required info (I called them pretty late). I googled around, and it seems Kaizen, the people who support Corsair in India, have an office in my city, and if various statements I saw here and there are to be believed, they have what is called an "express" replacement policy. I think I'll try them out tomorrow, if I can manage to get out of the office.
Thank you again, everybody, for all the suggestions and help!