If you have the software, this can be done easily in ArcGIS using the Zonal Statistics Tool in the Spatial Analyst Toolbox (http://resources.arcgis.com/en/help/main/10.1/index.html#//009z000000w7000000) 

I believe this can also be calculated in QGIS which is open-source, although I haven't used this.

If you have an ArcMap license, the tabulate area tool (found in Arc Toolbox under Spatial Analysis tools -> Zonal) can be used to calculate the area of each landcover type for each county.  The largest area would then be the predominant type.  If you don't have an ArcMap license, or are looking for an open source method, I've come across the LecoS plugin for QGIS: https://plugins.qgis.org/plugins/LecoS/ , and there's also a method using PostGIS: http://movingspatial.blogspot.com/2012/11/postgis-raster-tabulate-area.html . I will be trying to incorporate an open-source version of this in a project I am currently working on and will let you know if I come across any other insights; please keep me posted as well!

Using Zonal Statistics Tool in ArcGIS to calculate the area of each land use type. Then what you need to do is simply comparing those area values.