I know I only had a night to play Beta 3 so far (and what a night it was !), but I've still played it extensively enough to be able to provide some early Feedback.
I'm playing almost exclusively with Tarth, so of course, YMMV.
First, I must say I've seen major improvments in this beta. There's still a lot of work, but faction differentiation is starting to show up . It's really more interesting than beta 2, and there's a good foundation for an exceptional game.
That said, there's some issues I would like to bring up, and I'm sorry for not stopping after the congratulations. I'm not doing this to ruin everyone's day, but because I want FE to be a really good game.

-Balance:

There's quite a lot of balance work to be done. I really like the increase in power for non-damage spells (maybe healing should increase a bit too ; 8 hit points + 4 / shard isn't enough to have any impact on the outcome of a battle. 12 + 6 / shard would be a good amount, I believe.). That said, some damage spells are still way too strong. And in particular the usual suspects Fireball, Blizzard, Dirge of Ceresa and their per-soldier damage. The problem doesn't really lie within the spell, although they are a bit too good to begin with. The problem come from the combination of decent base damage with extreme scalability and low unit hit points.
The problem is that those spells don't just damage armies, they destroy them. The counter is very specific and costly, and needs to be applied to all units (spell/fire resistance!); Not to mention that a good spellcaster will shrug off spell resistance anyway.
In my estimation, an archmage (path of the mage, evoker III, 1 or 2 shards) killing half the soldiers in all units would already do a very, very good job. That'd be a very strong spell already. A beginner mage would kill 1 soldier per unit, or maybe 2 on larger, weak, unprotected units. That would already weaken the enemy army.
A reduction in power of the evoker series of traits (probably the most powerful by a fair bit), Path of the mage, a small nerf to the base damage and an increase in hit points would go a long way to make these spells less of a one-action game winners. (And yes, I know about counterspelling, but it's a very, very unreliable counter. You need a hero - you can't put heroes in every army - and it needs to be able to counterspell effectively.)
Another big issue is the extreme efficiency of armor. Granted, I was playing Gilden - but this just made things even more ridiculous.
I do not understand why armor values were increased so much. Armor was fine in beta .86. Sure, a fair bit of the damage still went through, but armor was a good mitigator. Now, it's a invulnerability shield.
I didn't even try to powergame - just to play the game normally. Basically, early in the game, once I dicovered leatherworking, I produced a single full-leather-clad unit of spearmen (with the trait that gives +3 armor on top of that), and casted stoneskin on it (for +15, since I had a shard). I attacked Magnar's capital with Marking and the spearman, plus a small unit of archers I had obtained along the way. It was a massacre; I must have lots less than 10 HP, against 9 groups of militia and spearmen. The issue was that my armored unit, even in enemy territory, was above 30 armor, making it almost invulnerable. I'm positive it could have won the battle alone. Later, I played another game where Marking reached 78 armor (and much more than that against blunt) in mid-game, again making him invulnerable to every non-magical weapon.
I understand that magic is supposed to help against such situations, but that doesn't remove the fact that armor values are way too high for now. Not all factions are strong in magic, and crystal isn't always abundant. It was said when War of Magic was in development that quality and quantity should both be viable. The way I steamrolled Magnar (and other factions) show that this goal hasn't been reached.
The problem with units being too fragile in .86 wasn't due to lack of armor, but of base hit points. The solution is thus simple: give units a base of 10 HP, instead of 4, and increase the power of all weapons accross the board. I've tested it: it works, and I can no longer steamroll Magnar... well, not as easily at least.
Also, dodge values are way too low. +3 dodge for a shield won't change much. 5 dodge against 3 HP? No thanks. Armor is way better than dodge, and dodge-related traits for units are too weak to be of any use.
Frankly, I wouldn't remove 3 HP from a unit (out of 8 in the unmodded game!) for 5% dodge. A unit with so few HP is going to die if someone sneezes too hard in the vicinity. It should at least have a very decent change to avoid damage. I would take that trait only if it gave at least 20 dodge.
Also, "+3 armor" and "+3 dodge, -10 carrying capacity" cost the same, but the first one is clearly much better. 3 dodge has a very limited effect, while 3 armor is quite a decent amount. 5 dodge, or maybe even 8, would be more appropriate to balance the cost and the penalty.

-AI:

The AI needs to learn to save money, research techs, and do something else than spawning sicty in every free square mile. The AI systematically discards quality over quantity. Not only is this counterproductive (because the game balance is strongly tipped towards quality), but it's also repetitive. I'd like to see some AI faction producing few soldiers, teching a lot, building a few strong cities, then maybe cranking out knights in full plates. Instead, all I see are militia and spearmen, with a few weak archers. The AI is painfully out-teched and out-gunned, and overexpands.
It should also learn to adapt to what I'm doing a bit more. If I'm going the heavy-armor way, it should produce magic-powered units. If I'm building a strong economy to produce high-level warrior, it should identify my production centers and destroy them first (for instance: iron mines...). If I seem to rely on heroes, it should switch to a hero-killing strategy. I would really like some SotS1-like adaptability (in SotS1, if you were producing drone ships or missile boats, the enemy would produce point defense vessels. If you were massing anti-capital ship weaponry, it would use light destroyers. etc...)

-Inconsistencies:

There's a few inconsistencies here and there, and in particular in pre-made unit design. I don't understand why the cenotaph has a trait that increase how much it can carry - all it carries is a spear, it'll never be encumbered. So I have to pay extra for a useless trait... many units have this problem.
The Gilden-specific warhammer becomes obsolete too quickly; There's only 10 turns between it and blacksmithing mid-game, and in my experience, it doesn't see much use. Why not decreasing the cost and weight of warhammers and Gilden-Warhammers, so that they will still see use when blacksmithing is discovered? Maces would be the strongest weapons, but warhammers could be a cheap, light choice for those who can't afford to wield their heavier brothers.