Chicken liver is the mildest that I've personally had.

Tongue. Tastes almost the same as muscle meat.

I made beef heart skewers a couple of months ago and they tasted surprisingly good. The meat is super-lean and needs to be marinated and cooked through, otherwise it's very chewy. You also have to make sure to trim off any of the remaining visceral fat and arteries, which calls for a very sharp knife and steady hands.

I cut mine into thin fajita-type strips and poured in lime, garlic, salt and pepper. Coated all of the meat with it and let it sit in the fridge for 24 hours. Then I slow cooked them over a grill until they were to medium, and turned up the heat until they got a bit of char. They tasted just like steak!

The second most mild-tasting organ is organic chicken liver. The regular livers you buy from the store from battery birds are strong-tasting and sort of nasty. Chicken livers are perfect for a pate (which calls for LOTS of butter!), and I also had good results by dredging them in egg and pecan meal, and frying in bacon grease.

Sweetbreads, wild deer liver (which is amazing, very mild but with super high retinol content atleast in finnish wild deer.). Whole cod liver, very nice texture, smooth and buttery. Beef pancreas might be too, i will get one next week so i will report back :)

Marrow is very mild, and so is beef tendon, which is nearly sweet. Lamb kidneys are suprisingly mild for a kidney, has to be fresh. Very nice in good mustard sauce, almost like some saugages :) Beef kidneys are never as good.

Calf liver would be milder, but at the end of the day it's going to be a strange taste and texture until you get used to it. So a good starting point is liverwurst and liver pates, any way of changing the texture and masking the taste so you get used to eating it and get enough confidence to just cook some with caramelised onions and wonder what you were doing missing out on this.

Tongue and heart, once processed, will be more familiar being muscles, but then that's kind of missing the point.

Lamb or beef heart. I regularly stir fry lamb hearts without guests even realizing it's not muscle meat. No prep or spices needed to cover flavor.

Lamb's fry (liver) is pretty mild. I cook onions in lamb fat until they're just starting to colour, and then I turn up the heat and sear strips of lamb's liver in the pan so they're browned on the outside but still well pink in the middle.

I had liver that's tasted unpleasant and I'm beginning to suspect that freshness plays an important part. Aging is good for steak, but I'm pretty sure liver should be as fresh as possible. Offal spoils fast.

Coming from the Midlands (UK) I was bought up on tripe. In fact, my Mum used to give it to me raw, cubed in malt vinegar and pepper. Deeeee-licious !!!! The other traditional way to eat it is with onions.

http://greatbritishkitchen.co.uk/recipebook/index.php?option=com_rapidrecipe&page=viewrecipe&recipe_id=1113&Itemid=28

I ordered a beef heart from US Wellness and it was very mild, as was the bison liver I ordered from them.

The best way to make these seem less strong is to cook them in fat that's one of your favorites and then, when it's about done, toss in a tbsp of water and cover it. The steam seems to take some of the flavor out.

And make sure you blend it with foods you like--bacon, favored veggies, etc.

Lamb kidneys are the mildest of the kidneys. I would also say it was milder than liver, but my tastes may be different from yours.

Calf's thymus is probably the mildest meat I have ever eaten and also very delicious. Its taste blends great with cream sauce.
Recently I've also tried pork spleen and it is probably the second mildest meat. Here's some good recipe:
http://www.nosetotailathome.com/2008/01/rolled-pigs-spleen/

Surprised no one is recommending duck. Duck organs are so universally recognized as delicious that they are often priced higher than duck muscle meat. I guess that's the problem... but duck organs are pretty readily available made into fancy pates.

What do the guts taste like? Sausage?