Idea #1 - CURVY
One possibility is to leave the straight lines behind and do the grid with curvy lines. Here are some examples.
|Tangles: Cubine, Eye-Wa, Florz, Juke|
|Tangles: Beadlines, Jemz, Knightsbridge, Pearlz|
Fragments: A3, and the 'Lisbon fragment'
Shave off alternating sides of the squares, like this:
Here's a tile with four examples. Sometimes it looks like tightly woven bands - like W2 without the black square 'holes'.
IDEA #3 - IRREGULAR
Another way to add variety and interest to a square grid is to vary the width of the stripes. Alternate a wide stripe with a narrow stripe, gradually change the widths, or be random.
|Irregular Beelight, Florz, and something I made up, I think.|
|Irregular Flukes, Zonked in the corners.|
There is a grid with offset large and small squares. It definitely takes some concentration to set up, but it's not too difficult and it's fun to work with. I used it in the tail of the peacock here.
Check out BrixBox from Anneke van Dam. Anneke's steps for setting up this grid pattern are well done and rather than reinventing the wheel I'll let you go to her blog to see it, as well as some lovely examples of ways to fill this grid pattern.
|Tangles: Cubine, Knightsbridge, Zewm|