There is no need to use the exact sizes in the instructions, but it should be close. Ideally the wood should be about 1/4" thick and 1 1/4" to 1 1/2" wide. The frames can be the same size or slightly thicker and wider.
The wood should not be too "hairy" the way some stickers that have been ripped from green wood can be. The frames look nice hairy though.
If your wood depot doesn't carry this wood cheap (15 cents or less a stick for 24" lengths), try snowfence, sandfence, lattice makers and some farming supply for stickers. Do not buy fancy wood for this project. You can use pine or any other fir tree and ceder is very nice.
If you have no luck, try wainscotting, either the front or back, or rip your own from a 2 by 4.
(I have used the backside of wainscotting that was double the required width (about 3 inches), but had a groove down the. It looked great and was very cost effective. For that project, I used some 1" by 2" wood depot wood for the frame.)