You can try looking for "Door stop molding" at the local home improvement store or lumber yard. It is usually 3/8 inch thick and 3/4 - 1 inch wide, and is generally in 8 foot lengths. If you have a small table saw you can cut it into strips about 3/16 inch square.
Another option is to buy some 3/16 square bass wood from the LHS, and make up a long piece by scarfing the ends and glueing them together.