Story: I have to give this series props for having an interesting story if nothing else. In my opinion, it was set up very well; the diaries, the game, the way the gods work (I'll leave it at that for fear of spoiling too much) all made for a very engaging and at least acceptably original story base. I'm sure the scry vs. scry idea has been the basis of at least a few stories in the past, but the way it was done in Mirai Nikki, with the whole cell phone diaries idea, made for some engaging games of wit and cunning since the owners' future predictions had unique and major limitations that kept every battle from being exactly the same. The rules of the game the plot revolves around were basically the bare minimum in the way of rules, so there isn't much to be said. It did a good job of moving the plot, but I think a god could've come up with something better. I also really liked the god's relationship with the universe, and the way gods in the series tended to behave. Normally I hate any supreme being aspect of a series, but I actually made a connection with this one. I've never seen a supreme being system QUITE like this one; it got a plus for me.
I also have to admit that I really liked the final eight-or-so episodes. But, once again for fear of spoiling anything, suffice to say they were done well.
However: Once we get into the more intricate details of the story, it begins to fall apart. The thing that most stuck out to me was that all of the emotions seemed really forced and jumpy. For example, if someone died, a character who witnessed the death would be upset for maybe a moment then get over it like it never happened. It genuinely seemed like everybody in the series had some sort of selective short term memory loss, or a Jesus-like capacity for forgiveness. (And on that same note, Hinata and Mao were seriously wounded several times and shrugged it off like it was nothing.) Another issue I had was the fact that Yuki and Yuno basically danced the same on/off, trust/no trust dance throughout the entire series, and their relationship waffled back and forth so frequently I began to get bored with their entire dynamic. And, perhaps most importantly for me, the beginning of the series felt really awkward. It seemed to me like everything was really rushed and contrived, like someone had listed the events and set a deadline for each one -- not that I doubt this actually happened, but a show isn't supposed to FEEL like that. Not only was there no real flow between events, but, in my opinion, there was not nearly enough time dedicated to important conflicts -- such as the battles with 9th and 4th -- as was necicary to give them the appropriate gravity, while very minor characters who probably should have been killed off in their debut episode or not showed up to Yuki at all, got the same time of day. The story also felt very contrived in that Yuki had an encounter with every single diary owner, and while I grant he is the main character and all, one could realistically expect that at least one or two would be killed off before they got to him, since the game is a free for all, not two against ten. And as I said above, that would leave more time to develop characters who actually mattered.
Basically, I liked it, but at no point did I become immersed in the story. I was fully aware of the fact that I was sitting at my computer watching a show the entire time.
Animation: I don't know, I really liked it. One thing that captivated me was the eyes; I've never seen that style before and something about it makes me really like it. Other than that, I just thought the whole art scheme was nifty. It wasn't super detailed or anything, and it's not like it would have taken a tremendous amount of work, but it had a subtle unique style to it I guess. As you can tell, I pay more attention to plot than aestetics.
Sound: Nothing really stuck out to me personally. I'm sure if you're reading this you may be thinking "Screw you, it's obvious the sound is amazingly well done because of reasons," or "Screw you, this series has the worst soundtrack in history becuase of words," but for me, the sound just got me through the show. I did, however, award two extra points to it for the intros, specifically the first one.
Characters: They didn't do anything for me. Yuno was the only slightly original and engaging personality in the whole series, for starters. With the exception of one I'd rather not spoil, Yuki and Yuno were the only characters who seemed to flow (and even that didn't make itself apparent until the last eight-or-so episodes). All of the other characters were either extremely flat or light switch dynamic (they were one way one second and then completely different the next). Once again, it seemed very contrived. I also was irked by many of the characters because of their aforementioned unrealistic capacity for forgiveness. Lastly, everybody cried too much, especially Yuki. There were a few times when I sympathized with them, but I usually just wanted to tell them to suck it up. I guess I just never really made a solid connection with any of the characters like I felt like I should have.
Overall: Despite my criticisms, at the end of the day I would recommend Mirai Nikki because it's a pretty interesting and original story, but I would tell the potential watcher to whom I was advocating not to expect a super memorable series. I enjoyed it all the way through, and while I thought it was of above average in quality, it wasn't particulary distinguished for me.