The first step to restoring travertine flooring is to clean it of dust and grime. Cleaning the tiles thoroughly requires the use of a tile and stone cleaner. Apply the cleaning product to the tiles, and let the solution sit for a few minutes, periodically brushing the dirtiest areas. Afterwards, wipe off the residue, rinse and then buff the tiles to a shine using a microfiber cloth.
To replace a loose or broken tile, drill into the grout around the tile in need of replacement. You need to drill the holes in succession to create a space wide enough to fit a knife. After drilling, pry the broken or loose tile off with the knife. If you intend to use the tile again, remove excess grout from the back of the tile with a multi-purpose razor scraper. Clean off the blank tile space using the same tool.
With the area cleaned, apply a thin coat of tile mortar or thinset to the blank tile space, then push the new or cleaned tile onto the surface. Use a twisting motion as you press the tile to for better adherence. Spread new grout on the sides of the tile, making sure to match the grout's color with the rest of the flooring. Allow the grout to dry, then wipe off any residue with a damp sponge. Repeat the process for every travertine tile in need of replacement.