Here is example code. It is fairly straight forward to adapt this code to work in a loop for processing all of your rasters. If your rasters share a common extent and resolution you can create a raster stack and loop through the bands in the stack. To create a vector containing all rasters in a directory, in a specific format, you can use "list.files" and then pass this vector to stack.
Example:
rlist=list.files(getwd(), pattern="img$", full.names=TRUE)
r <- stack(rlist)
# Add required libraries
require(raster)
require(sp)
require(rgdal)
# Set working directory, raster, in and out shapefiles
setwd("C:/test")
inshp="MyPolys"
outshp="PolyMeans"
rdata <- "Year2012.img"
# Read polygon feature class shapefile
sdata <- readOGR(dsn=getwd(), layer=inshp)
# Read raster
r <- raster(rdata)
# Extract raster values to list object
r.vals <- extract(r, sdata)
# Use list apply to calculate mean for each polygon
r.mean <- lapply(r.vals, FUN=mean)
# Join mean values to polygon data
sdata@data <- data.frame(sdata@data, m2012=r.mean)
# Write results
writeOGR(sdata, getwd(), outshp, driver="ESRI Shapefile", check_exists=TRUE,
overwrite_layer=TRUE)