How can I cut regularly spaced 45 degree grooves in a piece of wood?
You could use a router set in a router table, with a crosscut sled set at 45 degrees.



Building a jig with a "pin" that fits into the previously cut groove, will make sure all the grooves are cut an equal distance apart.  The "pin" will be attached to the sled in some way, and will sit against the edge of the work piece for the first cut.  After the first cut, the "pin" will sit in the groove produced by the first cut. The board is then lifted and moved, so that the "pin" sits in the most recently cut groove. This process is repeated the length of the work piece.

I'd make a jig similar to this.