I have a list in R with the following elements:
[[812]]
[1] \"\"             \"668\"          \"12345_s_at\" \"667\"          \"4.899777748\" 
[6] \"49.53333         
        In case you wanted to use the code you typed in your question, below is the fix:
listdata <- list(c("","668","12345_s_at","667", "4.899777748","49.53333333",
       "10.10930207", "1.598228663","5.087437057"),
     c("","376", "6789_at",  "375",  "4.899655078","136.3333333",
       "27.82508792", "2.20223398",  "5.087437057"),
     c("", "19265", "12351_s_at", "19264", "4.897730912",
       "889.3666667", "181.5874908","1.846451572","5.087437057" ))
v <- character() #creates empty character vector
list_len <- length(listdata)
for(i in 1:list_len)
    v <- c(v, listdata[[i]][3]) #fills the vector with list elements (not efficient, but works fine)
print(v)
[1] "12345_s_at" "6789_at"    "12351_s_at"
With purrr you can extract elements and ensure data type consistency:
library(purrr)
listdata <- list(c("","668","12345_s_at","667", "4.899777748","49.53333333",
       "10.10930207", "1.598228663","5.087437057"),
     c("","376", "6789_at",  "375",  "4.899655078","136.3333333",
       "27.82508792", "2.20223398",  "5.087437057"),
     c("", "19265", "12351_s_at", "19264", "4.897730912",
       "889.3666667", "181.5874908","1.846451572","5.087437057" ))
map_chr(listdata, 3)
## [1] "12345_s_at" "6789_at"    "12351_s_at"
There are other map_ functions that enforce the type consistency as well and a map_df() which can finally help end the do.call(rbind, …) madness.
If you want to extract the third element of each list element you can do:
List <- list(c(1:3), c(4:6), c(7:9))
lapply(List, '[[', 3)  # This returns a list with only the third element
unlist(lapply(List, '[[', 3)) # This returns a vector with the third element
Using your example and taking into account @GSee comment you can do:
yourList <- list(c("","668","12345_s_at","667", "4.899777748","49.53333333",
       "10.10930207", "1.598228663","5.087437057"),
     c("","376", "6789_at",  "375",  "4.899655078","136.3333333",
       "27.82508792", "2.20223398",  "5.087437057"),
     c("", "19265", "12351_s_at", "19264", "4.897730912",
       "889.3666667", "181.5874908","1.846451572","5.087437057" ))
sapply(yourList, '[[', 3)
[1] "12345_s_at" "6789_at"    "12351_s_at"
Next time you can provide some data using dput on a portion of your dataset so we can reproduce your problem easily.