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Don't use harsh chemicals for travertine cleaning including vinegar. Why because travertine is a very porous stone and can really end up damaging especially if its polished travertine. To clean the mildew off the travertine use a soft nylon bush with water and mild soap and of course using a sealer will prevent it to keep on coming back.
I would use dawn soap but use the normal blue kind or you may ruin the tile. Someone I worked for insisted I used vinegar and water and it totally messed up his tile. From that happening I have used Dawn soap and its never ruined any travertine I have cleaned. Hope this helps!!!!!