Utilty knifes cut strips cleanly.
Don't use power tools to cut thin strips of wood for crafts. Cutting thin, narrow pieces of wood on a table saw is dangerous. Hand tools are better. They can cut through wood cleaner and are much safer than power tools. Utility knives, pliers or even scissors will cut through most thin wood easily.
Video of the Day
Cut through wood up to 1/4 inch thick with a utility knife. This type of knife has a razor sharp blade. It also has a very narrow tip that will follow a straightedge closely. Just lay the straightedge on the wood. It's best if you clamp it to the wood, but you don't have to. Make an initial cut along the straightedge to score and form a line. The first cut should only be 1/32 inch deep. Make successive passes, each time going deeper until the wood cuts off cleanly.
Not typically used to cut wood, diagonal pliers will cut strips efficiently. They work like large scissors. They develop more torque than scissors though and will easily cut through plywood up to 1/4 inch thick. The detraction of diagonal plier strips is a rough cut. They can also smash the edges of the strip slightly. If perfection is not an issue, diagonal pliers can cut strips fast.
Scissors are one overlooked woodworking tool. There are industrial scissors that almost look like some kind of weapon. This large tool, sometimes over 12 inches in length, can cut through plywood up to 1/8 inch thick, or even hardwood if it is planed down to 1/8 inch. Scissors like these cut perfectly straight, with clean edges. The detraction could be that they take strength to use when cutting through thicker material. Gloves are recommended.
Veneer is thin, typically only 1/32 or less. It is real wood, used to glue over composite material to give it the appearance of solid wood. You can purchase almost any species of veneer already cut into strips. It is called wood tape. It comes in rolls like masking tape, but much larger. You can buy it with glue on one side or plain. You can cut it like paper using anything that you use to cut paper, including the old-fashioned guillotine-type paper cutters.