Yes, cosmic-ray astronomy is a thing, and it has been for a while.
More generally, there is multi-messenger astronomy since there are now multiple windows through which we are able to observe the universe via direct observation:
- electromagnetic (i.e., light rays, photons)
- cosmic rays (i.e. high-energy protons and atomic nuclei moving at relativistic speeds)
- gravitational radiation
Naturally, we built telescopes for observing light first. Galileo used an optical telescope and it turned out to be very fruitful for him, and it took off from there. Lens crafting became an art that was intimately tied with the progress of astronomy until the 20th century, where the discovery of quantum mechanics opened up the world of particle physics. Neutrinos sourced from nuclear reactions in the Sun, processes elsewhere in the Milky Way, and in other galaxies arrive at Earth and have been detected since the 1960s (although a large fraction of extra-galactic neutrinos are generated in supernovae, it is interesting to note that geoneutrinos are sourced from the decay of radionuclides in the Earth's interior, which helped to confirm the radiogenic theory of Earth's heating).
Over the past decade we have been able to begin doing neutrino astronomy, identifying distant neutrino sources.
Is there any reason that this cannot progress to other particles, I.E. general cosmic-ray astronomy?
There are numerous detectors all over the world designed to detect cosmic rays. There are direct (involving balloons or satellites in the atmosphere making measurements) and indirect (involving ground-based detectors searching for the shower of particles that cascade from the decay of cosmic rays in the atmosphere or searching for photons that are created from that cascade moving through the atmosphere) detection methods. I am generalizing, there are other methods and it is an active area of research (as Benny listed many examples in their comment).
For example, muons were originally discovered in cosmic rays shortly after antimatter was discovered. Muon detectors are of many types (i.e. Cherenkov-radiation water-tank detector) and can be used for all kinds of interesting purposes. For example, exploring the insides of pyramids and volcanoes without people getting hurt.
The recent direct detections of gravitational radiation by the LIGO/Virgo collaboration reveals that parts of the Universe we used to think we closed-off from observation are now see-able (or may soon be), for examples, at the early Universe prior to the cosmic-microwave-background radiation, and inside of compact objects such as white dwarves and neutron stars.
There could be other undiscovered windows into the Universe, for instance, if there is a "dark force" and thus a "dark photon" associated with dark matter, but that is speculative. We generally need reasons to suspect that something exists before we start making detectors to see them, but the early bird can get the worm, or sometimes it doesn't (like Joseph Weber!!).
Before the advent of large particle accelerators, i.e. 1950s, particle physicists relied on cosmic ray observations. These large accelerators, like at Fermilab or the LHC, are built to discover particles, and they have had varied success which also helped in the development of modern cosmic-ray detections that you could consider as being "general cosmic ray telescopes."