What's a good evergreen vine for Ohio?

Your planting zone is more important than your geographical ID. It looks like zone 5B in Cuyahoga county.

(http://www.touring-ohio.com/pdf/ohio-hardiness-zone-map.pdf)

Perennial vines for zone 5

Akebia (Akebia quinata)

Boston ivy (Parthenocissus tricuspidata) (You’ve ruled that out, I know).

Clematis (Clematis), selected varieties

Dutchman’s pipe (Aristolochia durior)

Trumpet creeper (Campsis radicans)

Wisteria (Wisteria sp.)

My personal favorite is clematis. It flowers, comes in many colors and sized flower shapes,
seed pods are attractive. You can also plant them in flowering sequence.