First of all, let me say (since that might not be immediately obvious) that overall I thoroughly enjoyed my return to Narnia. However, I recall various aspects I felt dubious about as a child, and my dubiety has grown with the years. On the whole they fall into two categories: sloppy world-making and the attitudes Lewis injects into the stories.
In contrast to Tolkien's decades-long crafting of his mythical history of Arda, Lewis seems to have made up Narnia as he went along, and sometimes you can see the join very clearly. One very obvious example is that, in The Lion, the Witch and the Wardrobe, it's strongly suggested that no humans have ever been known in Narnia before — Tumnus has a book called Is Man a Myth? In the later books, however, we're told that not only has Narnia always been ruled by humans before the Witch's reign, but there are plenty of other humans in the Narnian world, including those just over the border in Archenland. In fact, this retcon is implied even in the final chapter of The Lion, the Witch and the Wardrobe, where he refers to all the kings and princes who sought the hands of the adult Susan and Lucy. Presumably human ones.
Similarly, the layout of Narnia and its surrounding lands seems to shift about from book to book (sometimes even within the same book). For instance, in The Last Battle the young Calormene Emeth describes his home as lying Westward beyond the desert, whereas we know from The Horse and his Boy that the desert beyond which Calormen lies is to the south.
This sloppy world-making sometimes extends to practical issues. In the biopic film about Lewis Shadowlands, there's a scene where two children earnestly ask him where the Beavers got their food from, considering that it's been constant winter for a hundred years. Lewis has no answer, of course, and I doubt if he did in real life.
On the whole, Lewis just seems to have shoved whatever he wanted into his world. Tolkien criticised him for including Christmas into a world which had Aslan instead of Christ, and featuring Father Christmas there. Personally, it doesn't worry me too much — Christmas is just our current name for a very ancient feast, and Father Christmas is more than half a pagan figure — but I doubt that Lewis would have used those arguments as a defence. He includes just about every other mythical figure he could think of from our world, including the god Bacchus. Whether or not this was a good idea (I'm inclined to think it wasn't) the odd Christian myth is no more out of place than the others.
Lewis's attitudes are another matter. Some of these can be put down to his time and generation. By modern standards, the books can certainly be seen as somewhat racist and sexist — although, in fairness, female characters like Lucy, Jill and Aravis see more of the action than girls tended to in most children's books of the time. As far as race is concerned, the Calormenes come over as a very negative caricature of pretty much every "eastern" culture, from Arabic to Chinese.
A lot has been made of the Christian allegory in the series. I'll declare my position — I'm not a Christian, but I've never felt particularly anti-Christian, and mostly the religious allegory doesn't bother me. Whether or not it's also more, the Christian story is a great myth, and I've no problem with a story using its motifs, although I have to say that, as a child, my favourite books were always the middle ones, where the religious allegory took more of a back seat to the adventure.
The one exception to that is The Last Battle, where Lewis vehemently attacks the concept of different religions being essentially the same. Although he acknowledges that good people of other religions are "really" serving Aslan, he makes it quite clear that the Calormene god Tash is actually the devil. Tash resembles one of the scarier Hindu gods, though it's likely that Lewis was intending more of an attack on Islam. Whichever is the case, it doesn't do much for inter-faith understanding.
Lewis "shows" this by using one of his common approaches, of setting up straw men to knock down again. The ape Shift's arguments are so unconvincing (and we know they're false, anyway) that we're supposed to be persuaded that all such arguments are as phoney.
He uses this kind of technique both in Narnia and in our world. He makes Eustace and his parents thoroughly unpleasant, and then associates them with everything he disapproves of, from republicanism to wearing a special kind of underclothes. He criticises progressive education by portraying the kind of school very few people would defend, and his killer blow is that (shock horror) the Head is a woman.
The same thing can be found in the Narnian world. When Caspian confronts the Governor of the Lone Islands about the slave trade in The Voyage of the Dawn Treader, the Governor defends it in the name of progress, enabling Caspian to dismiss the whole concept of progress in a simile that would be easy to counter. It makes me wonder how many advocates of progress Lewis knew who would be anything but shocked by slavery.
Lewis can also be quite cruel. In Prince Caspian, Aslan punishes a bunch of obnoxious little boys by turning them into pigs, without any indication they'll be turned back. While it's true that children often enjoy this sort of draconian treatment of unpleasant characters, it hardly seems to fit the loving Christian message to condemn children to a terrible punishment for, essentially, being badly brought up.
Perhaps the most questionable message in the series comes at the very end. Philip Pullman has pointed out that Lewis is essentially telling the children reading the books that the sooner you die and don't have to live out your life in this world, the happier you'll be. While I've no problem with encouraging children (or adults) not to fear death, that's different from encouraging them to look forward to it. Enough kids attempt suicide as it is.
Well, you might be coming to the conclusion that I only have criticism for the Chronicles of Narnia, but, as I said at the beginning, I did thoroughly enjoy rereading them, with reservations. To turn from Lewis's weaknesses to his strengths, he always tells exciting and inventive stories. I said before that I always enjoyed most the books that concentrated on the adventures, without so much overt allegory, especially The Voyage of the Dawn Treader and The Horse and his Boy. The former is exuberantly inventive from start to finish, while the latter is a nail-biting adventure. On rereading, The Magician's Nephew has joined my favourites, although that contains more obvious allegory.
Perhaps Lewis's biggest asset, though, is his characterisation. The various children who visit Narnia are very realistic — even when they're nice, they quarrel and have moods, and they don't always behave ideally. Even Lucy (clearly Lewis's favourite) has her moments of weakness. On the other hand, he doesn't fall into the trap of making Eustace's transformation unrealistically instant, and in his subsequent appearances, when he's firmly in the "nice" camp, some traces of the old Eustace show through.
Even better than the children, though, is the string of Narnian characters they meet, especially the animals and mythical figures — Tumnus the faun, Reepicheep the mouse, Puddleglum the Marsh-wiggle, Bree and Hwin the horses, and many, many more. They're a sheer joy to read about.
And Aslan, of course. Allegory aside, Aslan's a wonderful character: strong and awesome, everyone's hope, but not always reliable as an instant answer. As the Narnians put it: He's not a tame lion.
One thing that did strike me very strongly on this reread was how Lewis's writing improved as the series progressed. This was brought home to me because I was reading the books in sequence, not in publication order, so I started with #6, #1 and #5. The fluent style and technique of the later books contrasted strongly with the awkwardness of The Lion, the Witch and the Wardrobe. The awkwardness has its charm at times, but not always. Does Lewis have to remind the reader quite so many times that it's foolish to shut yourself in a wardrobe?
The later books, although still in the chatty, storytelling style of the early ones, are much more assured, and on occasion they show ingenuity in their technique. In The Horse and his Boy, for instance, the hero Shasta gets himself involved in a battle. Quite realistically, he has no idea what's going on, so Lewis switches between what he sees of the battle and the commentary being given by the Hermit of the Southern Marches, watching far away in his magical pool. Without feeling at all contrived, it enables Lewis to give both an overview and an intimate immersion of the same battle.