Saturate some old rags or towels with water. Place the damp rags on top of the areas of caulk you wish to remove. Let the damp rags soak the caulk for 48 hours, which should soften the caulk and make it easier to remove.
Remove the wet rags and see if the caulk has softened. If it hasn't, dampen some rags with rubbing alcohol, then soak the caulk with the rags for 24 hours.
Lift the rags off the caulk. Wipe the caulk dry with a clean rag.
Slide the tip of a razor knife or a utility knife between the caulk and the floor. Gently wiggle the blade further down into the caulk and try to lift the edge of it out from the tile.
Grasp the end of the caulk with a pair of needle-nose pliers. Gently pull the caulk out from the joints. Repeat this step for each section of caulk you wish to remove.
Scrape out the caulk debris from the joints with the pointed edge of a painter's 5-in-1 scraping tool. Wipe out the joint with a rag or sponge.
Spray a mildew remover into the joints to kill any bacteria before you recaulk the joints. Allow the mildew remover to soak the joint for the time indicated on the instructions. Wipe out the joints before applying new caulk.