Thanks everyone for the excellent advice! I think I get the picture. So here's what I'm thinking, correct me for anything that sounds too out of whack:

1 - While there's still sap in the line, walk the line from bottom to top with a bucket of hypochlorite. At each tap, pull the tap and put it in the bucket of sanitizer. Once it draws enough to fill the drop and go a little into the 3/16 line, pinch the drop and put the tap into the cap on the tee.
2 - When I get to the highest tap, let it draw all the remaining sanitizer to fill the whole run of 3/16 line.
3 - Let it sit for a couple of hours. At this point all the drops and the whole line of 3/16 should be clean and full of sanitizer.
4 - Get a couple buckets of water (I figure you probably need more water to flush). Starting at the highest tap, pull the tap off the tee and put it in the bucket of water. Once the water column has rinsed all the way down to the next tree, plug the tap back into the tee, and go down to that next tree.
5 - Repeat step 4 until I get to the lowest tree in the line. On this one, let it draw all the remaining water through the line to rinse the 3/16. At this point, the whole line should be clean and rinsed.
6 - Un-cap all the taps. Does the order matter? Anyway, at this point, I think I now have a clean and dry(ish) line and drops.
7 - Plug the taps back into the tees until the start of the next season.

Yikes, if I've counted right, that means walking the line either three or four times (depending on the direction of step 6). Some of which is with heavy buckets. Did I mention the 20 degree slope? I'm going to earn a few beers that day!

Then on year 2:
- Before the start of the season, I walk the line, deal with any issues with tubing, replace all the taps.
- After the season, I do the same as season 1.

On year 3: instead of all of that cleaning, I just siphon sanitizer from the top tap all the way through the 3/16, and then repeat with water to flush it. And I need to cut out and replace all the drops and tees.

On year 5 I start from scratch, replacing everything.

And I thought cleaning a dozen buckets every year was a chore! You guys work hard for your hobby! LOL.

GO