Aside from scripting, you can do this with Automapping. You would set up rules that have the special “NonEmpty” tile as the input (i.e. match any tile) and “Empty” special tile as the output (i.e. delete that tile), both targeting the same layer, and use the Mod and Offset properties to make the rules only apply at the relevant locations. Since it looks like you want every second tile cleared in X and no skipping in Y, you’d need just one rule, and you’d set the ModX to 2, and OffsetX to 1, so that the rule only applies to every other tile in each row, starting with the second one.
You can even apply this automatically as you draw by using Automap While Drawing.
The Automapping documentation has been rewritten as of 1.9 and the Automapping system is easier to use starting with that version, so I highly recommend using a recent version of Tiled. Some of the features I described are not available in 1.8.2.