Most of your heart blocks don't appear too far off... Maybe If you can piece them by easing them together as close as you possibly can...after quilting and washing, it might not even be noticeable. ? ? Good luck.

For next time -
When one squares (evens up) the blocks - make sure that the point of the heart is "centered" -

For now,
I would add a solid black sashing - at least 1 inch (finished) wide -

The rainbow colored heart is the one that caught my eye - so that one would be the one I would use as a "focus"

This is what I was going to suggest. I think black narrow sashing would really make those hearts pop!!!

I have not read through to the end of this thread yet but this solution is what I was going to suggest as well. Fabric is forgiving. It does not appear in the photo that your blocks are off by that much. I think you should be able to 'stretch or squish' and get your heart sides to match up just fine. And even if they are off a smidge, by the time it's quilted it will never be noticed.

and once it is quilted, you really wont notice

Try sashing. This might even enhance the overall impact of your colorful hearts.

First of all, it's beautiful. Whatever fix you use, it will still be beautiful. I would try the easing, myself, and if that's not working, the sashing (but I'd use white, as I'm not fond of black, and I'd go wider).

If they are pretty close you could match the points on the hearts and adjust the seams where the blocks come together the other way to compensate.

I like the idea of off setting the columns. Move the second and fourth columns up by half a block, and take the odd half heart block off at the top. Then fill in those half block spaces at top and bottom with more black and white checkerboard fabric. I think the design would look more interesting than the lined up columns and rows, too. :o

Some block designs can't merely be "trimmed" without running into the problem you have. In those cases, I would make the block smaller by taking in seams, rather than cutting off at the edges.

I agree with the others about the 1" or even just a 3/4" sashing. That will merge everything together and make it work out. A second thought.....Your choice of background fabric is not one I would have chosen. I don't like it when squares don't match up, so it would drive me crazy trying to keep the integrity of the quilt pattern and the checkerboard fabric pattern. The sashing will help with that issue of it not matching up too. BTW...a very good effort there with your hearts. I like all the colors of the hearts and how you placed them. You have a good eye.