Decks, Patios, Porches and Docks - Help with finishing a pergola

I had a contractor build a large pergola over my 30'x15' deck. The rafters are 2x8 and are spaced approx 18" apart. I'd like to place some stringers perpendicular to the rafters for added support and additional shade. Any recommendation on how to go about this? I was originally thinking of just laying 2x4s across the top 18" or so apart and attaching to the rafters with decking screws.

I have a pergola over part of my deck. Rafters are 2x6 @ 24" o.c., from a ledger attached to the house, and extending 18" over a beam supported by long 6x6 railing posts. The rafters are trimmed on an angle about 12" back from the ends from the bottom up, leaving the end about 2x2. I put 2x2's spaced 2" apart perpendicular to the rafters, extending about 12" past the ends of rafters on both ends. I pre-drilled them to eliminate splitting, and attached each with a deck screw at each rafter.

Spacing, height, and width of the pieces is up to your own tastes, and how much shade you want. I have seen a number of sizes that I think look good. One detail I have seen that looks good, but seems like a lot of work, is to notch the boards so they overlap the top of the rafters - all the cross boards you are adding, times the number of rafters, can end up being a lot of notches.

I drilled mine from the top down, and set the screw flush with the surface. If you choose taller boards, (set upright) you might consider toe-nailing them (galvanized finish nails would work), as you probably don't want to use screws that long (expensive), or countersink them which would leave holes for water/debris to collect into.

Instead of 2x4s, use 2x2s spaced 2-4 inches apart. Drive a screw in at an angle, toe-nail fashion. I like this look better than 2x4s spaced 18". Overlap (sister) the 2x2s on a rafter.