If the look strikes your fancy - you could still go with the black using something that would "coordinate" with your blocks - maybe a pink, purple or teal. Imho, the black would really make them pop and the addition of some color in the plain blocks would soften the contrast. By the way - your blocks look fantastic. Nice job!!

What about, rather than a solid black for the alternate squares, you made square in a square blocks using fabrics from your pieced blocks as the inner square and black for the outside. This would bring out more color and also pick up the design of your pieced blocks as most of them have a square at center.

I agree with black - but be careful, as black can be very difficult to match! Can you contact the shop that ran the BOM to see what black they used? It will almost certainly be a different lot, but at least you could then use the same manufacturer. If that doesn't work, I'd use a sashing to avoid the blacks of the blocks and the blacks from the alternating squares. The one Betty shows above is gorgeous! It's a pretty thin sashing strip. Using a batik that has some of the colors of your blocks would work. Alternatively, I'd go with a gray or a color that you want to pick up from the blocks. If you go with a cream, be careful as you don't want the block on the bottom right to get lost.

Looks to me like grey would be a good choice. However, as with all colors there are many different greys. Do you have "stash"where you could try different colors. Usually you will find that you really like one --often one that you never thought of trying.

I think this is a great idea. The grunge black would give you black without being plain black and I think that would work well. I don't like the idea of using one of the colors in the blocks unless its a being or black. If you get too much color in the setting squares I'm afraid it would detract from the blocks themselves. They might even get lost in all the color.

I'd go with a black batik that has just a hint of multi-colors in it--like little dots or something. I don't like the tan or cream with this as your alternating blocks--distracts too much from your lovely blocks!

There's an aqua & green mottle that I'm liking. It's in the outer sawtooth points of the top row middle block and also in the inner star of the far right block in the second row. I would make alternate blocks of simple irish chain with this mottle for the background fabric and a dark for the chain squares. The black would be preferred but if there's hardly any left maybe just the center square of the chain blocks. Then I'd use a straight set so the chain blocks create the illusion of the quilt being set on point. I don't think a repeated alternate block would detract from the BOM blocks if done right. Same idea as eparys' quilt only with a twist. That's a well done quilt BTW.