Why don't we hear ultrasonic sounds as aliased versions of the original signals?
Short answer
The cochlea is a tonotopic map with certain physically determined boundaries that determine the range of frequencies perceived. Ultrasonic soundwaves simply do not have a correlate on this map. 

Background
The cochlea is a frequency analyzer that basically translates acoustic frequencies into a place-map. High frequencies are encoded basally (up to 20 kHz), low-frequencies apically (down to 20 Hz or so). Hence, it is pretty much a Fourier analyzing system (Fig. 1). This way of analyzing sounds is referred to as the place-coding theory of pitch. The place where a frequency is encoded is mainly dependent on physical characteristics of the basilar membrane in the cochlea. Every part is sensitive to a slightly different frequency then the next. This is caused by gradual variations in the stiffness and width of the basilar membrane, among other less important factors like hair cell length and so forth. The specific physical characteristics determine what specific resonant frequency a particular part of the basilar membrane has. Hence, incoming sounds are torn apart with standing waves, where each frequency results in a standing wave at a particular spot in the cochlea.   


Fig. 1. Tonotopic map of the inner ear. source: Ternopil State Medical University

The frequencies mentioned are physical wavelengths of the acoustic air pressure differences entering the outer and middle ear. The cochlea translates these into fluid-based pressure differences. Hair cels in the cochlea pick up these fluid pressure differences and tranlate them into potential gradient differences.

The sampling rate of hair cells is pretty much infinite, as they work on a continuous membrane voltage, i.e., they are analogue.

The secondary neurons, the spiral ganglion cells, translate these voltage differences into neural spikes and lead them through the auditory nerve to the brain.

Neural spiking follows the acoustic frequencies up to, say 1 kHz (frequency following response). This phenomenon is referred to as the temporal code of pitch hearing. After that, the refractory characteristics cause single fibers to fire only to one in a few wavelength periods. So at the upper limit of hearing, say 20 kHz, a ganglion cell may only fire once every 20 wavelengths or so. No problem, as many others do the same thing. Stochastics cause the wavelength to be nicely encoded in a population of responsive fibers. Furthermore, the auditory cortex contains a tonotopic map, meaning that high frequencies are processed elsewhere then lower frequencies. In other words, the auditory nerve doesn't need to faithfully encode the incoming wave. 

A nice example in this are cochlear implants; they stimulate the auditory nerve with electrical currents. The place of the electrodes determines the pitch, not their pulse rate (although it can have an effect). 

Now why are you not hearing ultrasounds? Simply because the basilar membrane does not contain regions sensitive to frequencies above 20 kHz or so. This is referred to as the Greenwood map, which depends on species.