How do you guys clean your silver ultegra rotors?
This question isn’t “how to clean disc brakes” - I can get the braking surface clean no problem.
I’m talking about the matte silver area inside of the braking surface. It always looks mucky and stained on my bike.
Maybe a good strategy by Shimano to sell more shiny black Dura Ace rotors 
I hear some folks are painting them shiny black to make them look as attractive as DA rotors at significantly less cost. Maybe try using high temp engine spray paint? My question is how will the paint react with the high temperatures in braking.
Painting them black doesn’t seem like a viable option for me - I’m not good with stuff like that and would also fear the paint coming off under heat.
I’ll postpone the DA rotor purchase to next year - so does anybody have an idea how to get the silver ones really clean?
You could use automotive brake caliper paint, and go black, red, yellow, whatever color. Typically good to 800-950F, should be way more than enough for a bike rotor. Often comes in a can for brushing rather than a spray can, so no worry about overspray. Whether it’s worth the material cost and time vs just buying a set of DA rotors though… shrug
…which is why I’d like to focus on the question on how to get the silver rotors clean 
Thanks for the tips, though!
mine look as good as new using warm water and dawn dish soap. These are on my gravel bike and get covered in EVERYTHING from mud to dust to cow crap.
if they are really stained, maybe try taking them off and use automotive brake cleaner.
Actual brake cleaner is highly toxic. Unless you’re dealing with contamination I’d stick with alcohol and a clean rag or paper towel, start at the outside and wipe towards the inside to avoid dragging any contam to the braking surface.
Red, you say? 
Funny you should bring this up. Yesterday after my ride I decided to do a general cleaning since I had the bike up on the stand to swap chains. I normally don’t clean the rotors but noticed they were a little grimy so I just wetted a rag with 91% rubbing alcohol and wiped down the mat surfaces. It seemed to remove all of the grime and as a bonus you can clean the braking surface if you happen to touch the rotors.
Water. If you’re nervous about that (and you really shouldn’t be, it’s not a chain), then alcohol. Applies to pretty much all non-drivetrain parts of your bike.
Then it seems I’m doing something wrong.
I tried soapy water with a sponge and I also tried disc brake cleaner - didn’t get them clean 
Scotch brite and a decent degreaser (do not dilute) will clean that up in short order.
Krud Kutter from Lowe’s works well.