Using North America Climate – Monthly Mean Precipitation data I was able to clip out my area of interest and use zonal statistics to produce a table of data for each county for each month of observation. While the dates of the dataset are from 1950 - 2000, I feel this is a good starting place for this particularly analysis.
Below is a video which shows the annual transitions using the original GIS data.
Below shows the general trends across Tennessee.
For a more detailed inspection, below you'll find the original maps for monthly analysis of precipitation in Tennessee. There are a few interesting things that show up when one looks at this data. October is clearly our driest month, which I found to be surprising. Precipitation is colinear with elevation to the point that certain geographic features are obvious in given months. One can clearly see the Blue Ridge at any time of year, and March begins to reveal the Cumberland Plateau and the Sequatchie Valley. The Nashville Basin is noticeable in all months but January.
Regional trends are also obvious, with the north-east Valley and Ridge province being consistently dryer than surrounding regions and the southern Eastern Highland Rim being consistently wetter than surrounding regions.

If you want to dig into the data and see what you can find out for your area, here is the table that I cooked out from the GIS data.