How to remove embedded dirt, etc from dimpled linoleum(no-wax)?

One cup of ammonia to one gallon hot water, a few squirts of Dawn dish liquid.

Ammonia might work. I found this from Armstrong. Keep in mind they are selling their own product!

Use Armstrong’s Once ’n Done or Multi-Surface Floor Cleaner Concentrate, but increase the strength, using up to three capfuls per gallon of water instead of the usual half-capful per gallon. (Armstrong also makes its Multi-Surface Floor Cleaner in a ready-to-use formula, labeled as a “refill,” and as a spray, but there is no way to increase the concentration in these.) Dribble the cleaning solution on tough spots and let it soak for two to three minutes, and then wipe the area with the sponge mop. For stubborn stains, scrub the spot with a soft brush or non-scratch nylon pad. And place a mat at all outside doors.

The product mentioned above is in stock at Home Depot. I remember cleaning a floor like that in a rental--not easy, but it did clean up.

I have used ammonia with a magic eraser & some elbow grease.

I have used Oxy Clean. Can make a spray with it or make a paste if the dirt is really imbedded. Allow the paste to set for a few and then use a soft scrub brush.

You're just going to have to get down there to scrub it with a brush, a little bit of dawn and peroxide. I've found that this combo is a great great cleaner for so many things. I swear by it.

I agree I've used lemon ammonia to clean embossed vinyl floors where the dirty water has settled into the design.

I also got best results using my carpet cleaner. It gets into the design and sucks all the water up. I put hot water, commercial carpet cleaner in the water tank.