Verify your island is one piece, no seams. If not ? You need more material. You don't want a seam in that.......

As to veining? they can show you the direction at the appointment. There's no way for us to tell you what/where /how based on a picture.

In that picture you showed though, I like the current position, because it hints at an upward movement. To the lower right, there is more grey like it has weight there at the bottom. Then that strong angular vein that goes from bottom left to upper right is a very strong lifting line. Visually it feels "right" and "natural". So whatever piece you have to work with, ask yourself if it feels like the pattern is "weighted" in a "natural" way  Does that make sense to you?

Now, from your question, I assumed you are talking about the backsplash. For the countertop, I guess it would help if you could look at it laying on the floor or a counter height and what you want to draw attention to. If your piece has a strong angular line like the one in your post, will it "point" to something you want to emphasize? Will it draw your eye towards something interesting? That's what I would ask. 

Definitely do as Jan suggests and ask if the piece for the counter will be cut up . Can you make a transparent paper cut out of your current countertop shape, cut in pieces or whole, so you can lay it, or them, over the whole piece in your appointment so you can see for yourself?

Do NOT put a seam on the island. If you need more material and budget is the issue? Tile the splash.

I probably would have changed the island configuration before new tops. ...........hard to say without more pictures of the entire kitchen.

Thank you! Very helpful. Out of curiosity, how would you reconfigure the island?

Reconfigure? You mean change the shape? If so, I can't imagine how you could improve it. Maybe if you explain what about it isn't working for the kitchen?

The L doesn't afford comfortable seating. We can't see the entire kitchen. We have no dimensions.

All I am saying, is .............before you make a cosmetic improvement of painting cabinets, changing tops, and all the effort it entails? You make certain you love the layout of all - appliances, storage, seating, etc.

Which is why...."Where is the rest" becomes the question. : ) Where is the fridge?? Someone, perhaps a previous owner, did some work before. ? Who knows? Too much info is missing to advise. It's ONE picture and a slab, that we are seeing here.