Wait until the agastache stems develop flower blooms and then begin trimming back the dead blooms daily or weekly to encourage new blooms. Use hand pruners to trim the stems 1/4 inch below the dead flowers.
Wait until fall months when frost begins to occur, and monitor the agastache plant stems until they turn brown. Some varieties die back immediately but others can last until winter.
Don gardening gloves, then gather all of the dead stems together and tie them into a clump with a piece of string. Position the string at least halfway up the stems and tight it tightly in a knot.
Cut the stems off 1 to 2 inches above the ground, using loppers or pruning shears and cutting straight across all of the stems.
Throw away or compost the bundle of stems.