I use Shaklee's "Basic H"
I went thru several products as well, trying to find a good cleaner without damaging the veneer. Murphy's oil soap doesn't work very well. Years ago I was a Shaklee distributor, and have used their product called "Basic H" for years for amost everything. It is a very mild cleaner and does not hurt the environment. It cleans by physical action, not chemical. I use about half-ounce of Basic H to a quart of water, and gently scrub the floor with a fairly soft brush, then just wipe up with a rag(tee-shirt). It removes the dirt very well, does not hurt the wood, then when it is dry, I just rub in Fornby's Lemon Oil. The result is terrific. Try it...You can get Basic H from any Shaklee distributor in any town. It cost about five bucks...but goes a long way and can be used for thousands of things. Kills insects, just spray on(diluted) and they suffocate. Makes grass grow better, etc....Good luck.