Ah! Easingwold - the centre of the universe?
I'm not too sure if the route I recently did is suitable for you "quicker" guys but after visiting Marston Moor, on the anniversary of the 1644 Civil War battle there earlier this month, I returned to near Easingwold via Whixley, Little Ouseburn, over Aldwark Toll Bridge (free to cyclists), Youlton, Alne (locally pronounced "Orn") and the minor road to Easingwold. The alternative to this minor road is the "shoulderless" A19 just to the east, scary, but some cyclists do it.
To get to Easingwold from Long Marston you have to cross the River Ouse at some point and this will either be via the Aldwark Toll Bridge, as already mentioned, or the York By-pass (also scary - for me) or going through the centre of York which you say you don't want to do.
Hope this is of some help.