For some reason, we have always played rolling initiative in each round of combat. It generates expectation to see who will go first on each round, as it can really turn the table. I can't remember if it was like this before, and the rules changed at some point, or how we started doing this, but I am curious to know if someone else rolls initiative like this, or if you see problems with it.

There are two points to it. We roll initiative each round, and we roll a single die for the players and another for the foes, then everyone adds their initiative modifiers.
In 2nd edition, that's the way it worked. Every time.

Now, what we used to do was 1d10 INIT + DEX bonus and it played out inverted, meaning the highest initiatives went last. What this simulated was that the people who were too slow couldn't react to the others around them, while the quick ones could. Damage was accrued each round after everyone had gone. If warrior A on INIT 4 hit warrior B for 20pts damage, but warrior B on INIT 12 killed warrior A, B essentially never took the damage because he ended up killing the slowed dude. Worked nicely, and it allowed really fast character to see what everyone else did and react accordingly, while the slow guys had to hope they were doing the right thing at the start of the round, although someone might "undo" they're action.

Startgate: SG1 RPG has a cool "fluid initiative" system that alters INT could (rolled once at the top of combat) depending on other actions and circumstances. If the character is caught in a blast, his initiative count gets worse, if he takes damage it gets worse, if it was a crit it's worse still. Aiming improves the initiative count, but aiming in SG1 is a full round action, so you get the count boost but have to wait unti lthe next round to take advantage of it, hopefully going first and blasting that baddie's brains out. Very cool system, and very likely exists in the Spycraft 2.0 system.
The rule in Conan (and d20 in general) is you roll initative once at the top of combat.

Since initative is cyclic and you are not flat-footed after your first round there is little benefit/penalty to changing your initative count. Rolling initative once speeds up combat since you don't have to recaculate everyone's place in the initative count each round but if you feel it is more "exciting" to roll each round then go nuts, it shouldn't unbalance the game.

We played White Wolf's WOD from the begining for a long time until recently, so Conan was our introduction to D20. I don't remember the mechanics of 2nd edition, but actually, I never figured ACs and to hits. More often than not I was the thief so I wasn't much into combat, because it was very frustrating, and it was more difficult to find the opportunity for a backstab than it is now for a sneak attack. I might reckon I had one of the players with the best initiative score being a highly dextrous thief, but fr what? I like the flat footed rules fine now. As I've said, it was long ago, and I don' recall those games for their mechanics, but for the good times.
:shrug: we like to get to the rolling of damage dice as fast as possible so anything that slows that down is un-fun.

To each his own.
Just in case, here's the fluid initiative modifiers from the SG1 game. Changing one's count over the course of a combat might make the difference in edging ahead of an enemy enough to take him out, so I can see value in it. Just for the record, too are some clarifications on rules: SG didn't differentiate between "move equivalent"and "standard", instead calling either "half actions". I've also left some out beecause they dealt with firearms and had no application in Conan.

+2 Aim (half action)
+1 Brace (half action) - hold your weapon against something to stead it for firing.
+1 move to higer ground
-- pass save to reduce blast damage to zero
-1 speed is reduced by terrain
-1 pass save to reduce blast damage to 1 or more
-2 character loses 1+ wounds*
-3 character becomes fatigued
-4 character uses weapon without proficiency**
-5 fail save to reduce blast damage
-5 character suffers a critical hit*

* First off, wounds were equal to CON and were only lost after all Vitality (essentially HP) were depleted (like in Modern). Secondly, these entries don't stack from the same injury.

** apply once in each combat for each weapon regardless of how many times the character uses that weapon.