I realize this question is old, but I feel compelled to add in here.
In addition to what's already been said (boost your MR, don't be afraid to over-nuke him to eat past his shield), there is the topic of range and escape abilities.
Mord is a unique melee in that his abilities have some good AOE and range to them, but he suffers the penalty of normal melee characters in that a good ranged will still be able to keep him at bay. Characters with long range and/or skill shot abilities like Anivia, Ezreal, Kog'maw or Karthus can put pressure on Mord very easily during the laning phase, which is where Mord tends to establish himself. A character with strong lane presence/harassing ability like Nasus, Garen, Vladimir or Fiddlesticks can also hurt Mord's early game by keeping him at the enemy tower where the tower will eat most of the kills.
Mord also suffers from a distinct lack of escape abilities. Ganking Mord can be easy with a properly built tank to bait his abilities into cooldown, and then bum rushing him when he has nothing left with which to counter. Slows and stuns prevent his otherwise relatively high HP and shield from making a difference, so plan ahead.
Mord tends to be very item dependent. A tank Mord ends up having lower overall damage, while a DPS Mord ends up having a very low lifepool and no escape mechanisms. Putting pressure on him in his lane early game prevents him from racking up too much money (which is why he was designed with so much AOE ability), and thus prevents him from getting his really good items at the proper time which would otherwise allow him to become the "powerhouse" that everyone fears.
Don't get me wrong, a properly played Mord is quite good, and a smart player knows how to balance between him being a tank and a burst DPS. But a heavily pressured Mord will crumble late game, and a ganked Mord will have no chance against the proper lineup.
Tryndamere is one of League of Legend's melee carry-s, which means, indeed, that he becomes nigh-unstoppable lategame as he gets more and more items (though despite what you may think, Tryndamere is not nearly as powerful now as he was back near release)
Tryndamere's counter is hard CC. If you don't have CC (which is quite possible in blind pick), you have no real option other than to prevent him from getting kills in the first place.
This is, in part, why Tryndamere is considered something of a "newbie stomper" - without CC and coordination, both often lacking in newer players, he can be a royal pain to deal with.
The truth of the matter is that Tryndamere, like all carry-s, is fragile. Without his ult, he melts like paper, especially under focus.
What Tryndamere wants you to do, more than anything else, is stand still and let him crit you to death. So don't let him - use your CC to snare him, slow him, keep him at arms length. (I dare say you'd be hard pressed to find a team of 5 without one slow or better betwixt all of them)
The goal when fighting Tryndamere should be to get him to use his ult, then disable him for its duration, then kill him. If he uses his ult to flee, all the better - re-engage on a team fight within 2 minutes, and it'll still be on cooldown, making him prime to focus.
In terms of surviving Tryndamere, you want armor, and lots of it. A Thornmail will cause Tryndamere's massive crits to likewise sliver his own health, while a Frozen Heart will reduce his attack speed while providing you armor (and mana).
Best Answer
As Riven, your biggest advantage is your mobility. Use your abilities to hit them a few times and get out before they can significantly trade back. This will work for most juggernauts, but there are some that will be able to trade back if they hit an ability (such as Darius). Against them, you'll want to either bait out their abilities before going in to trade, or out-play them. Using Darius as an example, if you are hit with his E try to stun him while he is casting his Q, hit him and use Q or E to get out of range. Depending on the matchup and relative skill, in some cases you will have to basically do what you are saying you do and just get good at farming under turret until your jungler comes.
With Nasus, you are signing up for a weak early lane in exchange for quite a lot of power once you've stacked your Q. Focus on minimizing damage taken and stacking at first, get some damage if you are doing OK or some defense and CDR if you aren't, and start using your Q to keep your opponent wary of trading with you. As Nasus going even in a lane is basically winning it (assuming you are stacking), and use your E to make trades go in your favor once your Q is sufficiently powerful (by hitting them with Q, then slowing them to keep them in range for a second one).