You can do more than tabby (plain weave) with a rigid heddle. You start with Pick Up Control. That's done by pushing your heddle into the down position and inserting a warp separator such as a stick, paper, cardboard or anything into the shed created behind the heddle so that you can see only all the slot threads. Then you use a pick up stick to pick up only those slot threads you want to use in your design. Once you do that you remove the separator and leave only the pick up stick in place.
Once the stick is in place, it's a matter of leaving it near the warp beam (neutral position) or bringing it behind the heddle, either laying flat or turned on edge.
In this video I picked up 2 slot threads, skipped 2 slot threads, then did a repeated sequence of moving the heddle up or down, while moving the stick forward or back, etc.
Here's how to produce a waffle weave (great for towels or blankets)
U = Heddle up
D = Heddle down
U+S = Heddle Up and Stick forward (flat)
SE = Heddle neutral with stick forward but turned on edge
Pick Up Control: Skip 2, Pick Up 2.
U, SE, U, SE, U,
D, U+S, D, U+S, D
Another Waffle Weave:
Pick Up Control: Skip 1, Pick Up 1
D, U+S, D, U, SE, U
Don't have a pick up stick? Next time you or someone you know makes a visit to the hardware store, have them ask for a couple of wooden paint stir sticks.
You can use a ruler or stiff cardboard too. Look around the house, you're bound to find something that will work!
Here's a wonderful video by Kelly Casanova showing how to do waffle weave.