After you stitch all the horizontal rows, clip at each intersection and alternate the seams (just like you would do if you were doing a regular simple 9-patch).
Now sew all the vertical rows, just taking care to keep the seam allowances in the correct position.
After the first step, I normally "split" that horizontal seam and press it open carefully. Much less bulk to deal with.
Did you cut the squares 1.5" for a finished "postage stamp" of 1"? Or did you cut them 1" hoping to finish with..what?
If you've miscalculated, you can save it all by sewing with a 1/8" seam, but sew carefully, reduce your stitch length a bit, and make sure your horizontal and vertical seam allowances are all consistent.
And, yes, that's why viewing postage stamp quilts always results in "was she crazy?" comments. Because of the thousands and thousands of squares.