Flagstone tessellations are one of my favorite variations on geometric origami designs. They are easy to solve, but difficult to fold.  This particular design was conceived of by Madonna Yoder (aka gatheringfolds). Essentially, it's just alternating rhombuses.  As with most flagstone tessellations, the reverse side is a fairly dense field of triangle twists.  I created a similar flagstone some time ago that was also all rhombuses. Mine all went in the same direction, while hers are more of a back and forth pattern.  Her design creates a really cool wave pattern. It's a brilliant find.  To fold it you'll need a lot of patience. Flagstones are fussy to execute.  You can see the triangle pattern that's created on the back in the second photo. The back side is almost more impressive than the front.  Crease pattern is pretty straightforward. See the image below. 

  This is technically not a tessellation. The pattern doesn't repeat. I fiddled around a little bit with the possibility of it repeating on a larger grid, but didn't find a suitable solution.  That's not to definitively say that it can't be repeated. Perhaps it can. I just didn't find a way. Maybe if you change out the first outer rhombus where the central rhombuses meet and work with a shape that is more natural to the corner that is formed. Or maybe a third rhombus on the outer edge.  I didn't really try that hard, since if possible, it would require a very large grid.  It is, however, a pretty cool pattern that can be folded from a 32 pleat triangle grid.  During the conception process, I just came up with the central pattern and from there free formed the remaining edges.  It's a neat little dense fold that isn't too hard to execute.  A crease pattern is included at the end of this post. 

 Conceiving of this tessellation I was remembering one by Arseniy K where he used the same central module. I added some blunted pyramids to the configuration to create a different pattern. In this case, I decided to have the center be latent and every other external repetition dominant.  The remaining repetitions are hybrid latent and dominant.  But this particular tessellation can be altered with how you choose to orient the folds. What goes up and what goes down has a lot of flexibility.   .  Crease pattern is included at the end. 

 This tessellation was created by gatheringfolds. It's one of her less complicated designs. An array of trapezoids arranged linearly.  She folded hers using a rectangle.  I went with a hexagon.  Even though it's very simple, I really like the arrangement.  Sometimes it's fun to just figure out something not too complicated and quickly get on with folding it.  The trick to solving this is just to fold two of the kissing trapezoids. Then  follow their axes. They will tell you where to go from there. You also need to be aware of spacing. Too close and it won't work. Too far apart and you'll wind up with a different pattern.  The reverse side has a pretty cool design as well.  Crease pattern is included below. 

This is a fun tessellation of trapezoids repeating on both the front and the back.  The trapezoids alternate amongst each other between front and rear facing.  Completing that trapezoid pattern results in some large hexagon twists.  There are also a multitude of triangle twists on the back side.  This is one of those patterns where you have choices about how to orient the folds.  What goes on the front and what goes on the back is personal choice. Some of those choices mean that other folds are mandatory in a certain direction. But you can mess around with the pattern to some extent.  I guess technically, that's true of a lot of tessellations. But some patterns present more choices than others.  I believe I drew a crease pattern, but I left it at work. Will try to add it at a later date.  Update: added crease pattern.

 This tessellation is by Arseniy K. It's one of his more straightforward designs.  It's pretty cool still the same.  Not a whole heck of a lot of solving on my part. It's a series of repeating small rectangles around triangle twists.  It is difficult to fold, even though it's not that complicated. Everything is quite tight.  I kept staring at it after I'd already completed it and wondering if there was something that I had missed.  Try as I might, even with fancy paper, I could not get my folds as neat as his.  If there is some subtle difference in how to execute, I'm unable to see it.  He only posted a photo of the front side. So I can't compare my reverse side to his. As far as I can tell the fronts look the same.  The one drawback to really fancy, thick origami paper is that it doesn't generally back light well.  In this case, the front didn't back light nicely at all, but for some reason the back view actually came out p...

 Here's another interesting linear style tessellation created from rhombus twists. It consists of rhombuses twisted in horizontal rows. It's repeated vertically by tucking the resulting intersections backward.  I forgot to take a photo of the crease pattern and I don't have the paper on which I drew it anymore. Duh on me.  But I will try to remember to draw a new one and upload it here sometime soon. It's a pretty simple design though. Not too tricky to figure out how it works.  I used a hexagon to fold it, but upon looking at it more carefully, I imagine a regular rectangle would've suited this particular configuration a little bit better.  The pattern on the back side is pretty cool as well. Nothing terribly intricate, but I like it.  It's always nice when everything tucks and nestles together so neatly.  Update: made a crease pattern. Added it below.