Say the data was in Row 1.

This will go from A to D to G ... etc.

Enter this formula anywhere, and copy down:

=INDEX(Sheet1!$1:$1,3*ROWS($1:1)-2)

How do you transpose every third cell in a row of data to a column on a
separate worksheet.  There should be no empty cells in the column of the
separate worksheet that this information is being copied to.

Try adding another column which has an X, say next to every third row.
This can be done by putting an X next to the first row, selecting three
cells and dragging down to the end of the data. Then apply
Data>Filter>Autofilter and select X from the new column. Now you should
be able to copy and select Edit>Paste Special>Transpose on the new
sheet.