Oh boy, do you have a tough situation. I adore travertine but, even sealed, it can absorb stains pretty easily. I use a Bissel Steam Mop to clean mine and a Bissel Broom Vac for vacuuming. LOVE my Bissel broom vacs! You'll need a natural stone cleaner for in-between steam cleanings. Natural stone requires a neutral pH balanced cleaner. I use Aqua Mix Natural Stone Cleaner (not positive that's the exact name). You can buy stone poultices at Lowes or Home Depot to remove stains. Of course steam is the best and safest way to clean your tile floors. An alcohol and water mixture is safe to use on natural stone and inexpensive. I use it on my granite countertops.
Monica

Thanks a bunch! Perfect ideas!