Why is the brain dependent on glucose?
I want to present another (possibly more practical) approach towards this phenomenon. Lets begin with amino acids as alternatives.


Amino acids, apart from being a source of toxic ammonia, are also harmful to the brain in another way. The two (concerned) types of amino acids, aromatic and acidic ones, are a lot more than just energy sources for the brain. Aromatic amino acids: tyrosine, phenylalanine and tryptophan, are precursors for the biosynthesis of neurotransmitters like serotonin, melatonin, norepinephrine, dopamine, etc. The greater the amount of precursors, more the product formed. Talking about acidic amino acids, glutamate and aspartate are themselves neurotransmitters. So, same case could apply to them too. So, it might have been that neurons, in evolution, decided to just get rid of such sensitive and powerful energy sources which can disrupt their basic function. You can see Fernstrom, 1994 for a study on this.
Fatty acids are definitely a nice candidate. But they have some side effects too. For a long time, it was thought that since fatty acids (or lipids in general) are attached to albumin while in blood, they cannot cross the blood-brain barrier (see Stryer for example). But, this is not the case, as now proven. Then why are fatty acids not a preferred energy source for brain? There might be quite a few reasons for this. 


First, $\beta$-oxidation of fatty acids demands more oxygen as compared to glucose, making neurons more vulnerable to hypoxia. For example, the reaction for complete oxidation of glucose is (from Molecular Cell Biology):


$\ce{C_6H_{12}O_6 + 6 O_2 + 36 ADP^{3-} + 36 P_i^{2-} + 36 H^+ \rightarrow 6 CO_2 + 36 ATP^{4-} + 42 H_2O}$

On the other hand, the reaction for $\beta$-oxidation of palmitic acid is (from pharmaxchange):

$\ce{C_{37}H_{66}N_7O_{17}P_3S (Palmitoyl-coA) + 23 O_2 + 108 P_i^{2-} + 108 ADP^{3-} \rightarrow C_{21}H_{36}N_7O_{16}P_3S (coenzyme-A) + 108 ATP^{4-} + 16 CO_2 + 23 H_2O}$

where palmitic acid is $\ce{C_{16}H_{32}O_2}$. This clearly shows that glucose oxidation requires less oxygen (1 molecule per carbon) as compared to fatty acids (almost 3 molecules per 2 carbon).


Second, nonesterified fatty acids (NEFA) decrease membrane potential at inner mitochondrial membrane, causing collapse of electrochemical proton gradient. 
NEFA also interfere with electron transport chain, stimulating generation of reactive oxygen species like superoxide ions (see Zhang et al, 2006 for example).
Apart from this, the rate of ATP generation from fatty acids is slower as compared to glucose, which makes glucose a better option for neurons (see Stryer).




All these factors might have a played a role in making neurons choosy for glucose. This is also supported by the fact that one of the enzymes of $\beta$-oxidation, 3-ketoacyl coenzyme-A thiolase, has significantly lower activity in neurons as compared to other tissues (Yang et al, 1987). You can see Reiser et al, 2013 for a study on this.


PS: It has been shown that the brain can also use lactate along with ketone bodies as an energy source. See Wyss et al, 2011 to know more about this.