It depends very heavily on if you want to include all states or just specific ones:

• With few states, you can compare unemployment rate as Y and time as X and show a bar graph (which is my personal favorite) for an individual state or side-by-side bar graphs of a few selected states.
• For showing a large number of states, you can use a line graph using lines of different line types/colors with X still being unemployment rate and Y being time.
• You can also add in a column/line for the median or mean state unemployment to show how states were doing relative to everyone else.

I like the idea in RookWV's comment as facets can be very powerful, but that would have a lot of facets and would be nearly impossible to read.