Thanks to this article on doing the heavy lifting.
In Ubuntu:
RUN apt-get update \
&& DEBIAN_FRONTEND=noninteractive apt-get install -y --no-install-recommends \
tzdata \
&& apt-get clean \
&& rm -rf /var/lib/apt/lists/*
RUN echo America/Chicago > /etc/timezone \
&& ln -sf /usr/share/zoneinfo/America/Chicago /etc/localtime \
&& dpkg-reconfigure -f noninteractive tzdata
In Alpine, I'll have to do more experimentation:
RUN apk --no-cache update && \
apk --no-cache upgrade && \
apk --no-cache add tzdata openntpd