Above is the before picture of my floor under my fridge. I hate it too, but with the right tools you can see that from the below picture it cleaned up very well.

Above is the after picture from the fridge floor clean up.

The materials I used you ask???

1. White Vinegar
2. Baking Soda
3. Spray bottle
4. Scrub Brush
5. Wash Rag
6. Water
7. Bucket

Step 1. Take the vinegar and put in the spray bottle, DO NOT dilute the vinegar.
Step 2. Spray vinegar on area you wish to clean.
Step 3. Sprinkle baking soda over sprayed area, and take spray bottle, spritz again over the baking soda.
Step 4. Let stand for about 5 min or so, then with a bucket of hot water take your scrub brush and scrub area.
Step 5. Take wash rag and wipe excess up with it. Get a clean bucket of about 3 cups of vinegar and 1 cup of water. Use the rag and and vinegar/water and wipe floor again this will take off any of the left over baking soda the vinegar with bubble it off and you will be left with a sparkling floor.

This is a great cleaner, it also disinfects and deodorizes the floor area for a fresh feeling. This is a non toxic approach for a cleaning alternative. You may have to tweak it a little bit like I did to your liking. Just as a reminder always test before you use anything in a area you done use or under things before using it on main area. I hate my lanoilium (not sure on spelling it wouldn't find it for me) floor in my kitchen, bathroom and entry way; so if it makes a boo boo I don't care because eventually I will have a new hardwood floor in all those areas. Good luck with all of your cleaning. ;)