Problem: Amazon is rolling out a feature for Alexa where a landlord can use it to collect rent payments, thereby tying the Alexa experience to the renting experience in ways that are less easy to escape.
Bigger problem: Amazon is also rolling out a feature where someone who owns the Alexa can pair the two devices so they can use one to listen in on the other, thereby creating even more privacy intrusions.
Nice solution: Build an easily-hideable faraday cage for the Alexa by lining a cardboard box in many layers of aluminum foil. Trust me, it works, and I know from personal experience, because the metal-foil part of the insulation in my parents' house blocks all cell signals.
Nasty solution: Play very loud music, which will be sweet revenge upon the landlord, and will either force them to reveal that they have been listening in, or force them to stop.
Very nasty solution: Have loud sex right in front of the Alexa.