There are a few difficulties in building Neutrino detectors. First and foremost, Neutrinos only rarely interact with ordinary matter. For a neutrino to have an effect on ordinary matter, it has to strike the nucleus of an atom. Unfortunately, atomic nuclei are ridiculously small in comparison to the atom itself - by diameter, the atomic nuclei is about a million times smaller than the size of the atom itself. By volume, it's around one billion billion (10 to the 18th power) times smaller.
Neutrinos hit solid chunks of matter about less often than a particle of light strikes a piece of dust in a nebulae in space. It happens, but you need huge amounts of matter for it to happen. Nebulae are light-years across, with only a few motes of dust every meter. If you could somehow see neutrinos, the earth would look like a particularly faint, spherical, nebulae (if it could be seen at all).
If you built an extremely long detector (preferably hundreds of meters), you would detect neutrinos on rare occasion. But because the rate of neutrino detection depends on how much matter you are observing, and less on the arrangement of that matter, it'd just be more efficient to build a giant sphere of matter, carefully measured for neutrinos.
This brings us to the second problem. When a neutrino hits a piece of matter, it releases a small amount of energy. A neutrino detector is mostly just a giant chunk of matter, and some detectors that can sense those small bursts of energy. But lots of other things also cause small bursts of energy. The only way around this is to block everything that isn't a neutrino. Thankfully, this is relatively simple. Pretty much everything blocks anything that isn't a neutrino. That's why they often build neutrino detectors in salt mines or other underground, highly insulated places: you could spend millions of dollars building a huge concrete pile of stuff to block 99.9999999% of everything that isn't neutrinos, or you could just built it underground.
What you're proposing is possible (linear detector), although you couldn't fit it on a desk...