I would rather that the floor is at the same height.

When was the house built? An older Dutch Colonial would have had all hardwood floors, possibly excluding the kitchen.

Previous owners may have tampered with the original flooring? I am going to guess the dining room floor is the culprit.
I would love to hear how floor joists could run in different directions.

There is a tool for locating joists,,,called a stud finder and it will beep or flash when it locates.

Lay the new living room floor perpindicular to the joists and rip up the dining room floor and lay it the same.

Run them perpendicular. In a traditional floorplan like that, you can even set off the living room with a border. That will make the differences seem intentional. But even if you don't, it really shouldn't be a problem. Having the floor perpendicular might even mask any mismatches in the two differently aged floors.

It would not bother me in the least. We have that setup.

Floor heights are to be maintained as much as possible (safety reasons as well as annoyance). An option would be to match the SPECIES of the wood you have in the other room, lay it in the correct direction (perpendicular to the joists) and then stain it a DIFFERENT colour. It can be tone on tone. That way you have a species match that, in the future, can have a colour match when you choose to have the floors refinished all at once.

Of course you could add a decorative "in-lay" border in the room and then run the planks perpendicular to the joists. The border adds visual interest as well as a raison d'etre for the new floor to run in a different direction. You can make it look like was done on purpose.

A solid, site finished hardwood installed by a craftsman installer can get this done in a heartbeat. The beauty of the inlay makes the up-charge worth while.

Perpendicular and level. With a rug in the living room, you probably wont even notice.

Being kind of OCD I would probably run it in the same direction as the dining room and use saddles on both rooms (even though one room doesn't need it). If you decide to run the wood perpendicular, you can buy an area rug for the dining room that hides a lot of the floor.

Judy G, I will tell you how joists run in different directions. Our house was built in 1990. We have a fairly open floor plan with a walkout basement that mimics the rooms upstairs in most circumstances. We have engineered joists that span the large open room ( 24 foot wide). Where this room meets up with the rest of the house there is a large steel beam. The rest of the joists run the other way off of the beam. Floor plan done by well know in our state architect and reviewed by structural engineers.

We had this issue when we put down hardwood and installed another layer of sub floor so the wood could run all the same direction.

I would not want different heights. It would look patched together, to me. And would be a tripping hazard. (Even one inch would be a hazard. I'm a klutz.)

Replace the plank subfloor with regular 4 x 8 subfloor panels. And run the flooring the same direction as the existing.

Thanks guys; this was really helpful! I think we're going to stick with the subfloor as is and just run the wood perpendicular.

FWIW, in my den remodel/addition, I had to run our hardwoods perpendicular to the flooring in the next room. Seriously thought it would make me crazy. I honestly don't even notice it now that the room is complete. I mean, it's there. I just don't think about it, because it doesn't stick out like I thought it would.