It's been a little while since I dealt with time-aware data in ArcGIS but I don't think it's capable of handling time-series data when the different time periods are stored that way, with each time period as a separate attribute of one feature. Instead, you will need one feature per county, per time period. ArcGIS time animations require a start and end time for each feature, so your feature class' attribute table would look something like this:
County Unemployment DateStart DateEnd
Monroe 3.4 1/1/2010 1/31/2010
Monroe 3.5 2/1/2010 2/28/2010
Monroe 3.8 3/1/2010 3/31/2010
Logan 4.2 1/1/2010 1/31/2010
Logan 4.5 2/1/2010 2/28/2010
Logan 4.4 3/1/2010 3/31/2010
Then when you use those features in a time animation, each row will only be visible between its start and end dates.
If you just need to export a set of maps, each one using a different field for the symbology, you can do that with Python using the
arcpy.mapping module by looping through each of your time period fields and changing the
valueField property on the symbology class each time. See these questions for more on that:
How do I change graduated color class values with python?
Accessing "valueField" property of layer's symbology using ArcPy?
And ESRI's help on changing the symbology using