It has a regular expression feature for replacing. You can use a regex that matches #region as well as whatever else is on the line, and replace it with empty space.
Search menu -> Find... -> Mark tab -> Find what: your search text, check Bookmark Line, then Mark All. This will bookmark all the lines with the search term, you'll see the blue circles in the margin.
Then Search menu -> Bookmark -> Remove Bookmarked Lines. This will delete all the bookmarked lines.
You can also use a regex to search. This method won't result in a blank line like John's and will actually delete the line.
Older Versions
Search menu -> Find... -> Find what: your search text, check Bookmark Line and click Find All.
Then Search -> Bookmark -> Remove Bookmarked Lines
Jacob's reply to John T works perfectly to delete the whole line, and you can Find in Files with that. Make sure to check "Regular expression" at bottom.
Using regex and find&replace, you can delete all the lines containing #region without leaving empty lines.
Because for some reason Ray's method didn't work on my machine I searched for (.*#region.*\n)|(\n.*#region.*) and left the replace box empty.
That regex ensures that the if #region is found on the first line, the ending newline is deleted, and if it is found on the last line the preceding newline is deleted.
Still, Ray's solution is the better one if it works for you.