Ah... the lovely Axe-Kick.
If you can see the Axe-kick coming - best thing to do is the X-block - cross your arms above you head and block the kick. From here if his kick gets stuck on your guard then push him backwards while following up with a HARD low kick.
Or if you want you can just 'accidentaly' trip him as you push him back. Falling from a blocked Axe-Kick will do some nice damage... and bruise his ego.
If he knows you can repel the fancy stuff then he wont bother trying it.
Most axe-kicks Ive seen in amateur fights are very clumsy and rather useless. Just to get a pop out of the crowd really. Never seen anyone get knocked down or even hurt by one of those. One fighter used it excessively against a friend of mine and it seemed to be irritating him more than anything else.