The basic idea is that there are two groups in conflict while searching for the route to Yggdrasil - the Maginians, which you start out working with, and the Seafarers (the people "descended from the Vikings" that I mentioned before), who have opposed each other for generations. For the first few dungeons you're just searching for routes further along the world map, and as you go you're learning more about some ancient civilisation who used to inhabit the area (the plot reason for the dungeons being reused from past games is because this civilisation created their own Yggdrasil using the energies of the other trees around the world).
When you reach the Petal Bridge, you meet Artelinde (without Wilhelm) and Wulfgar Jr. (who as the name implies is descended from Wulfgar from EO2, so it looks like Hrothgar and Wulfgar living is probably canon now, rip the Juggernaut theory) - who are working for the Seafarers, and basically tell you to get lost. You obviously ignore them like any good masochist/explorer would, and press on anyway. In one area of the Petal Bridge, there's a fight against three or four sets of monsters in a row, and at the end of it you find the remains of some Seafarer soldiers, and their valuables with a request to return them home. So presumably the end result of all this is that you end up caught between the two groups, and you either have to choose one to stick with, or help both of them to reconcile. I like stuff like this, so I'm interested to see where it goes from here.
(It reminds me somewhat of the plot of EO3, based on what I know about that game, which had the player choosing between Armoroad and the Deep City. Plus some of EO4 which had a lost civilisation that you only hear about for much of the game before eventually coming into contact with them and their remains.)
Meanwhile as all of this is going on, there's some other stuff about the Maginian princess falling ill, and they make a point of telling you how unusual this is, so you probably get charged with figuring out what's wrong with her. And some dude who just randomly shows up out of nowhere and never gives his name or anything - he acts all polite and helpful, but I bet he turns out to be working for some enemy power or other, so he'll probably end up being the game's human antagonist.
That's as far as I've gotten for now, anyway. Overall it seems like they're borrowing ideas from EO3 and 4, which are usually considered to have the strongest stories in the series, and mixing them together (so like everything else in the game). It's still not overly deep or complex or anything (which is how I think the series' stories should be, anyway), but it's got a bit more going on than most other games and seems to be setting up the player's party to actively drive parts of the plot too; usually the player is more of a passive participant in the plot.
So I like what I've seen so far, but I should probably reserve judgement until I get further along in the game, in case they completely drop the ball and the whole Seafarer plot is over and done with in the space of a single dungeon.
