Need a COUNTIF-like function to count rows, not cells

I'm looking for a formula that will return to me the number of rows in which a particular set of text appears in at least one column. For example:
Example Data - Colors.PNG

Each row represents an article, each of which has three sections. Each section includes one or more colors. I want to know how many articles include the color green in at least one section (i.e. in how many rows does the text "green" appear at least once).

This part was easy: =COUNTIF(B2:D7,"*Green*")
But of course it returns the number of cells, not the number of rows.
The color green appears in article 1, 2, 4, and 6, so the answer I'm looking for is 4.

I'm sure there's an easy way to do it - any help is very much appreciated!
Thanks in advance.

Thanks so much! That will certainly work. Much appreciated.