Becoming 6th dan in tenhou boosted my confident to teach basic strategy quite a bit. Not as much as it would if I could enter houou tables, but still quite a bit. Today will not be one of "xkime's translations" but rather "an article actually completely wrote by xkime." Enjoy.
The theme is: The order for discarding honors.
Starting from the beginning, why should we discard them? Lone honors are the tiles most unlikely to become mentsu (groups). Number tiles have from two to five ways to become mentsu, while honor tiles only have one: drawing two of the remaining three. You may have been told by Janki movies or by Tsuchida Koushou to never discard honors in your first discard, but restricting yourself with this rule is absurd.
You may be a supporter of "shibori" ("squeezing," keeping the tiles other people may want to prevent them from completing their hands). This "shibori" strategy just backfires most of the time. You slow down your hand a few turns to keep that Hatsu, then someone else just draws it and discards it and blam, it gets ponned by someone. What a waste. You reduced your amount of useful tiles in vain. Not only that, but that person may even draw the last one for an ankou, or completely give up on it and aim for a higher scoring hand (a game with players who shibori tend to go a lot more into late game, it seems, where hands can get pretty expensive...). Another reason to avoid shibori is, today's mahjong is about speed and atozuke (back-door win) is pretty usual. It would be of just pretty bad taste if some guy would just pon and chii everything from every other guy, and once you got to tenpai (or near tenpai) you decided to discard that yakuhai and dealt into his hand. (The basic of shibori is, if you're going to shibori, you must shibori until the end!). It's pretty usual that your hand improved to some point, and you're left with a dangerous live honor tile. There is no meaning in "live tiles" in early game, so unless you have an otherwise very important reason to keep them, discard them then!
Some other people believe that any honor (among fellow yakuhai or otherwise) are the same. They are not. There is an efficient discard order. They are all very different, and so their value changes. Given that none of them are discarded, you want to cut the following tiles:
What's the best order in every case? Let's watch it case by case.
East Seat in East Round.
North, West and South are not yakuhai, so they are useless. You'd probably want to discard them anyway, and if they do get pon'd by their respective "owner" it's best to start with North. He is the player to your left, even if he calls the tile, you get an extra draw right away. Next is West, because it skips South. South for last, as it changes nothing. Then the sangenpai (dragon tiles); you always want to discard sangenpai before your seat wind tile because sangenpai can be used by everyone while your seat wind is most useful to you. The most efficient order for sangenpai is Haku, Chun, Hatsu. There is actually not much of a difference between Haku and Chun, but according to Mii's research, saving Hatsu for last is somehow tied to a very slightly higher hand win rate (it surely is a better discard order than "haku hatsu chun" or "chun hatsu haku" and a lot better than doing it randomly). If you always follow that system, it will be alright for the most part. Then, at last, your Double East. You may want to insist on this tile the longest, as it grants you two han points instead of one.
South Seat in East Round
Why is the round win in first place? The current mahjong strategy is, discard from the tile you don't want to get pon'd! If the south player discards East in his first discard, the chances it gets called by East are extremely low. You can pass on that danger there. Even if it gets called by East, he is to your left so you get one more draw. Next, North (it skips west player), then West. As explained above, Haku Chun Hatsu. Then in the end, the tile only useful to you, South.
West Seat in East Round
East, same reason as above. South, as it skips north. Finally, North. Haku Chun Hatsu as always. Then at last your own wind, West.
North Seat in East Round
(You may leave East for one turn or so later if someone already discarded it, and jump to the next live tile)
First we cut off the double east's poison. Then West (it skips south). South. Haku Chun Hatsu. Finally, our North.
East Seat in South Round
(The truth is, if you need yakuhai badly, you can leave South for just before Haku Chun Hatsu in all of these except South's case)
The case you want to avoid the most is South Player getting Double South. Starting from there, you then go on with North (skips south and west), then West (skips south). Haku Chun Hatsu. Then the tile only useful to you, East.
South Seat in South Round
Starting from East. It's useless and it would be more expensive for East to have it. Also, even if it gets called, you can draw an extra tile right away. Then North (skips west). West. Haku Chun Hatsu. Then, at the very last, your Double South.
West Seat in South Round
Avoiding south's Double South as much as possible (at least 2000 pts), then east's East (1500pts). You may invert this, but the truth is, even if South calls it, you get an extra draw right away. Then North. Haku Chun Hatsu. As always, in last, your seat wind.
North Seat in South Round
From the dreadful Double South. Even if it gets called, the Dealer in East gets skipped. Then East. West. Haku Chun Hatsu. In the end, no surprise, your seat wind.
With this thought pattern you can even deduce yourself what's best in the rare West rounds or North rounds.
Also, the order may be affected by the amount of discarded tiles, or your hand. For hon itsu, you want to increase your chances of yakuhai tiles as much as possible (leave all yakuhai for last), while for chii toitsu and pinfu (or hands that are already pretty much ready and you don't want to pile up yakuhai as much as you want your opponents not to call them) you want to concentrate on non-yakuhai tiles the most (start discarding from yakuhai). But using this basic scale of values is the entrance to a good discard order.
Oh, also, don't forget to check the dora.