What’s the best way to build a fence on hilly ground? If we put the posts in the ground perpendicular to the boards it saves a lot of cutting and fitting. Also the gates will swing without hitting the ground (or leaving a big gap under the opposite end).
We can see more reasons to set the posts perpendicular to the ground than to set them plumb. Are we missing anything important here? 
What kind of fence? You’ve mentioned posts and boards, but what method of construction? Are the boards nailed to one side of the posts? If so, then having the posts vertical, and the boards parallel to the ground sounds like the better compromise to me.
A gate that’s hung on a vertical pole has the benefit of not having to fight gravity as it’s opened and closed. Depending on the length (and consequent heft) of the gate, this can be a significant factor.
Seems like it’d be simpler to build a gate that’s somewhat triangular in shape to accommodate the terrain.
Tilted fence posts sound like a problem. I can’t say they are for sure because I’ve never seen it done. You would assume the horizontal sections would hold the tilted posts in place but I think you are will just have the whole weight of the fence pulling the posts down toward the ground, eventually weakening the whole thing.
The usual solution I see for this is to stair-step the fencing up the slope. At the bottom of each section you’d put some fill in between the lowest horizontal fence board and the ground.
Trinopus and Mangetout have mentioned the best options. Take a look at these images of fences. Planning a fence like this will require staking out a line so that you can see exactly how much fall there is and where the steps need to be, if you opt for steps.
Those are some pleasing photos of fences!
As others have said, the only way I could imagine a fence being constructed with posts parallel to the ground is some kind of rough log fence - but even w/ split rail, I imagine the posts will be plumb.
The standard practice with board on board/picket style fences is to rack the sections so that the cross pieces are at an angle, but the uprights are vertical. The alternative is to step it, keeping the crosspieces horizontal, but varying the length of the uprights.
You definitely want the gates to be hung from vertical posts.
Take a bit and try to google images/discussions of fences installed otherwise. Drive around the area with similar terrain and see what others have done. I’d be surprised if you turn much up.
Posts not set plumb are going to look super weird and janky (that’s an official term). And I’d think they’d be structurally compromised as well.
Okay then… thanks for all that! I’m glad that I came here before we started cutting up expensive lumber and having to dig up and reset the posts!!!
We lived in Indiana and hills weren’t any consequence there. (The only hills in Indiana are the exit ramps.) But now we live where noting is flat - you even need to chock the grocery cart at Kroger’s to keep it from rolling away, striking other cars when you turn your back. (Why don’t they put parking brakes on shopping carts?)
New environs bring new problems - or - at least problems not encountered before in our world. So thanks for the information and advice. That gate with the offset hinge is worthy. They say any day you learn something, the day isn’t wasted.
If you set the posts plumb the force of gravity on the whole fence will push straight down the posts. If you set them perpendicular to the ground gravity will be pushing on them sideways, which opens up more opportunities for instability.