We've got honed travertine and its an absolute nightmare. Recently had them professionally cleaned and the difference is amazing. I have no doubt it will need to be done relatively frequently, but its worth it because they're back to looking like new again. For ongoing maintenance I'm using white vinegar and a bit of eucalyptus oil for mopping, after vacuuming them very thoroughly first.

I've tried a few steam mops and I really don't think they make a difference - maybe professional grade ones do but I can't afford one of them ;-)