Synthetic biology has successfully advanced our ability to design and implement complex, time-varying genetic circuits to control the expression of recombinant proteins. However, these circuits typically require the production of regulatory genes whose only purpose is to coordinate expression of other genes. When designing very small genetic constructs, such as viral genomes, we may want to avoid introducing such auxiliary gene products while nevertheless encoding complex expression dynamics. To this end, here we demonstrate that varying only the placement and strengths of promoters, terminators, and RNase cleavage sites in a computational model of a bacteriophage genome is sufficient to achieve solutions to a variety of basic gene expression patterns. We discover these genetic solutions by computationally evolving genomes to reproduce desired gene expression time-course data. Our approach shows that non-trivial patterns can be evolved, including patterns where the relative ordering of genes by abundance changes over time. We find that some patterns are easier to evolve than others, and comparable expression patterns can be achieved via different genetic architectures. Our work opens up a novel avenue to genome engineering via fine-tuning the balance of gene expression and gene degradation rates.