What are disasters with Axiom of Determinacy? It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. Herrlich's book on AC has a chapter aptly named Disasters with Choice.

But omitting Axiom of Choice and working in ZF alone might lead to even more weird things. (Again, just have a glance at the table of contents of the same book and check various things mentioned in the chapter Disaster without Choice.)

Therefore there have been some suggestions on possible replacements of AC. For example, for some things working with ZF+countable choice is enough.

One of possible candidates is Axiom of Determinacy. For example, it is known that AD implies
countable choice for sets of real numbers. (See, for example Theorem 9.29 in L. Bukovský: The Structure of the Real Line.) Again, for some purposes this version of AC might be sufficient. AD also implies that every subset of $\mathbb R$ is Lebesgue measurable.



Let me quote from the book Pavol Zlatoš: 
Ani matematika si nemôže byť istá sama sebou - Úvahy o množinách
nekonečne, paradoxoch a Gödelových vetách; (Even mathematics cannot be certain about itself, Essays on sets, infinity, paradoxes, and Gödel’s theorems, in Slovak), page 117.

The (not very good) translation is mine. This excerpt follows after mentioning French school represented by R. Baire, E. Borel and H. Lebesgue and Russian school lead by N. N. Luzin.


  Both these schools substantially used set-theoretical methods in analysis, topology, theory of functions, measure theory, etc.,
  but Zermelo's proof of well-orderability of continuum and some other "unpleasant" consequences of Axiom of Choice disconcerted them to the extent
  that they challenged this axiom and ineffective existential proofs in general.
  
  Let us mention that if Axiom of Determinacy had been known at the time, it is possible - or even probable - that these four great mathematicians would have given preference to this axiom over Axiom of Choice. Using Axiom of Determinacy it is possible to build nicer, more elegant and unified descriptive set theory, where various pathological (e.g., non-measurable) subsets of real numbers are impossible.


Or the quote at the beginning of the chapter on AD in H. Herrlich's book (which is attributed to U. Felgner and K. Schulz):


  Among all alternatives to the axiom of choice AC the axiom of determinateness AD undoubtedly is the most interesting.




I wonder whether working in ZF+AD would also lead to some consequences which some people might consider undesirable.


  
  What are consequences of Axiom of Determinacy which might seem antiintuitive/problematic/paradoxical?
It might not be a disaster, but I've always thought it counterintuitive that there is an equivalence relation on the real line with strictly more than continuum many equivalence classes.  The equivalence relation isn't even very complicated; it's just congruence modulo $\mathbb Q$ in the additive group of $\mathbb R$.  In other words: Vitali wants to give you his standard example of a non-measurable set, and, if you prevent him from doing so, he gets his revenge with this example of partitioning a set into strictly more pieces than it has elements.
You need AC to prove Gödel's completeness theorem, and that is a big deal.

See the wikipedia article on Completeness Theorem which I cannot make link.