I just finished all four scenarios included in Arkham Horror 3rd edition. These impressions are written from the point of view of someone who has never played any previous edition of Arkham Horror or Eldritch Horror or Arkham Horror: the drug game. I am generally pleased with it but not blown away. It felt like a more exciting version of Pandemic.
Some issues: it takes many hours to play a scenario. Around three, I think? I always needed to take a break halfway through, and sometimes finished the scenario on the next day. This is because it takes time to gather the clues that are usually required, and the investigators often need to be elsewhere, trying to reverse the 'anomalies'. Even though it has a strong storyline, the game feels very mechanical. Your main concern is to eliminate the doom tokens that will cause an 'outbreak', if left unchecked.
For some reason, you are not allowed to perform the same action twice which doesn't make any thematic sense. You begin to attack a monster, but suddenly have to stop. You want to move further but can only take two steps unless you pay (for a cab, presumably), and move two more spaces. Then you are forced to do something else, and the only remaining option may be to gain a dollar.
Monsters feel more like hindrances than real monsters. If you have a strong investigator, it's not difficult to take them down. The encounter cards are the heart and soul of the game, but there is only 8 of them in each neighbourhood. Of course we are talking FFG here, so why would they provide expansion material in the base game? At least the flavour text is very well written, and makes the game worth playing. As for randomness, it is quite strong, but, for me, not annoying. The encounters shower the characters with items, allies, spells etc., so it doesn't feel unfair.
All in all, I'm happy I played it, and will probably pick up the expansion later. The rules are extremely simple, and the game is immersive. I would only recommend it to people who are not enamored with previous Arkhams and Eldritches (they may find it disappointing by comparison), and have not fallen into the sinkhole of the card game. The only problem is that it made me curious if I would like the card game better. I'm trying to resist finding out...