Dig out and hand-pull individual plants. Look for the center crown, and hand-pull the plant by grasping the main stem close to the ground and pulling. Dig resistant specimens out with a trowel or shovel. Cover the area with a thick layer of organic mulch, such as shredded bark or wood chips, to keep any stray seeds from germinating.
Cut the plants down to ground level, using a pruning clippers or mow them down with a lawn mower. Then cover the area with several layers of cardboard or newspaper to keep light from reaching the stubs of creeping thyme. Anchor the cardboard or newspaper to the ground with rocks or stiff wire bent into loops. Cover the area with a thick layer of organic mulch. Check the layers of cardboard or papers periodically and replace them if they deteriorate. Leave the area covered for six to 12 months to eradicate the thyme.
Spray the creeping thyme with non-selective herbicide, such as glyphosate. Perform this on a calm day so the spray does not land on desired vegetation. The herbicide will travel from the plants' leaves into their roots, killing them within 10 to 14 days. Remove the plant carcasses when they are dried and brown.