Consider a transmission line shunted at some point with a diode and an ac pulse (a wave packet, say) propagating along it. When the pulse reaches the diode, if the diode is forward-biased, the wave packet will see a short and will reflect back, inverted. On the other hand, if the diode is reverse-biased, the wave packet will see an open circuit (with perhaps some extra capacitance, which can be mostly compensated for by reducing the distributed impedance of the transmission line or by adding inductance) and will pass through unimpeded.
Perhaps by this means you can compute the bitwise OR of two bitstreams stored in a transmission line with a diode, by looking at the signal past the diode; and perhaps you can calculated their bitwise AND by isolating the reflected signal.
Off-the-shelf PIN diodes like the US$2.73 (US$1.27 in quantity) M/A-Com MADP-011027-14150T operate at up to 12GHz. So you could imagine doing this operation on bits that were 13mm long traveling through a transmission line at 0.5c.
With an ordinary diode, unfortunately, this provides no way to amplify the signal.
Specialty parts like Gunn diodes (available as USSR new-old-stock on eBay for US$8–20) can operate at higher frequencies, up to some 40GHz in some cases (supposedly up to 200GHz in some devices I haven’t seen for sale), and additionally have a negative differential resistance region which can serve to amplify signals. Supposedly, gallium nitride Gunn diodes can reach 3 THz. (Tunnel diodes, which are also specialty parts, are also a possible option, although they do not reach such high frequencies; they are available on eBay at much lower costs, like US$1.)
These frequencies are far, far higher than any transistor.
Suppose that you found a way to do universal computation with some kind of network of transmission lines and 3 THz Gunn diodes. You could very reasonably use 100 meters of coaxial transmission lines in a desktop-sized device. At 3 THz and a transmission speed of 0.5c, the transmission lines would contain about 200 million oscillations at any given time, each about 50 microns long. You could imagine this waveform containing 400 million bits. With a reasonable number of bitwise computing elements, such as 32, the device would perform 96 trillion bit operations per second. A Skylake CPU with three 256-bit arithmetic units might perform 1536 useful bit operations per cycle (if we count an add with carry as two bit operations), which would be 6.144 trillion bit operations per second at 4GHz. So such a device could be computationally useful even without integrated circuits.