df2 <- df1[Reduce(`&`,lapply(df1[paste0('year',1:4)],
           function(x) !x %in% c('B', 'C', 'D', 'U'))),]
df2
#   year1 year2 year3 year4 signup_date
#5                              8/3/08
#6     A                        4/6/07

df1 <- structure(list(year1 = c("B", "C", "U", "B", "", "A"),
year2 = c("U",
"D", "C", "", "", ""), year3 = c("C", "B", "D", "", "", ""),
year4 = c("D", "U", "U", "", "", ""), signup_date = c("4/10/12",
"2/12/12", "3/14/05", "3/7/05", "8/3/08", "4/6/07")),
.Names =   c("year1",
"year2", "year3", "year4", "signup_date"), class = "data.frame",
row.names = c(NA, -6L))

library(dplyr)
filterVars <- c("B","C","D","U")
df %>%
  filter(!year1 %in% filterVars, !year2 %in% filterVars, !year3 %in% filterVars, !year4 %in% filterVars) %>%
  arrange(desc(signup_date))

year1 year2 year3 year4 signup_date
1  <NA>  <NA>  <NA>  <NA>      8/3/08
2     A  <NA>  <NA>  <NA>      4/6/07