Vinegar will certainly surprise you, as it's acidic and will soften and etch your lovely travertine, which is mainly calcium carbonate. And you don't have to blindly trust an anonymous poster telling you the registered user who told you to use vinegar is wrong -- use Google. A mild base (i.e. Alkaline) will be much safer, and if you've ever used bleach in a shower you know already how effective that is on mildew. But I would start with something much milder still, like a dilute hydrogen peroxide.
