Pour 1 part bleach and 4 parts water into a bowl or small bucket. Insert the blades of a pair of pruning shears into the solution to disinfect them. Soak the blades for five minutes. Let the blades air-dry before making any cuts.
Put on a pair of gloves to protect your hands from injury. Select a water sprout with green flexible wood and a height of 6 to 12 inches or less. Grasp the sprout in your dominate hand just above its base. Place your non-dominant hand on the main branch from which the sprout originates, holding it in place. Pull quickly and firmly on the water sprout while bending it severely to one side to snap it from the branch. Repeat this process with all remaining softwood water sprouts.
Select a young water sprout, 2 to 3 inches or less in height. Place your thumb at the point where the sprout's base joins with the tree's main branch. Push your thumb firmly against the sprouts base to rub or snap it from the tree. Remove all similar water sprouts that remain too short to grasp and pull off the dwarf peach tree in this same manner.
Cut through the bases of any mature, stiff water sprouts with hardened wood using the pruning shears. Make the cut as flush with the sprout's parent branch as possible without cutting or damaging its bark. Do not try to pull these hardwood sprouts from the peach tree as it may split or tear the parent branch's bark.