Monotony is not just about the tone of your voice staying at the same level. Any aspect of your voice, including the cadence/speed of your speaking, can become monotonous. Repetition of any aspect, whether speed, pitch, tone, volume, and the words you use can be boring or hypnotic--and we don't normally want to either bore our audiences or put them in a trance.
One quick way to practice vocal variety is to pick up a children's book and read the story out loud. You can do it with your kids, or even with adults.
There's just something about reading a children's story that brings out the storyteller in just about everyone. It brings out our vocal variety.
Audio record it! Listen to what you did with the telling of the tale, and how your voice maneuvered through the process.
If you don't want to use a children's book, I can recommend the book I've used in many of my classes and workshops: Fractured Fairy Tales by A.J. Jacobs. Yes, this is a book collecting many of the Jay Ward Fractured Fairy Tales that appeared on the Rocky and Bullwinkle shows.
My own vocal variety comes from watching Captain James T. Kirk's monologues on the original Star Trek. I often tell people "I went to the William Shatner School of Speech!" Whatever one might say about Shatner's acting, he is one of the best examples of vocal variety out there.