I sympathize with @Noiselab as my 2009 Mac Pro stopped working a couple months ago with the exact same symptoms: No chime, dead USB, no way to get it to do anything.
The Mac Pro had its firmware and CPU upgraded several years ago and had been working just fine, then one day it didn't. All board LEDs look normal. For reference at the time of failure, it had these specs:
4.1 > 5.1, 6-core 3.2Ghz (W3670), 32GB RAM, AMD Radeon 570, macOS Mojave 10.14.6, PCI SSD.
I've pulled out all the parts that I could (HD, RAM, GPU), swapped in others, and even ordered a new power supply from eBay, but nothing changed the behavior. It's not worth it to me to try to replace either logic boards as I had already switched from this to a Mac mini i7 as my primary machine last year.
Now it's a mostly empty shell is sitting next to my desk where it serves a slightly uncomfortable guest seat. I hope you have better luck figuring out what's wrong with yours.
Thank you for your kind response. I hope so too! You know how it feels...
Time is precious and is sometimes running out too fast for me to afford the luxury of troubleshooting around a lot ...
BTW if it's not actually as empty as it sounds... you can still try to power it on once more...
Those things often behave in... mysterious ways
Anyhow regardless of all this mess I have to admit that never before in history there was so much info and knowledge available to anyone at the cost of a click... for anyone who wants to...
I'll see what I can do about it, and shall update the thread anyway.
Cheers