One issue with https://stackoverflow.com/a/16144262/2055486 is it assumes neither data frame already has duplicated rows. The following function removes that limitation and also works with arbitrary user defined columns in x or y.
The implementation uses a similar idea to the implementation of duplicated.data.frame in concatenating the columns together with a separator. duplicated.data.frame uses "\r", which can cause collisions if the entries have embedded "\r" characters. This uses the ASCII record separator "\30" which will have a much lower chance of appearing in input data.
    
setdiff.data.frame <- function(x, y,
    by = intersect(names(x), names(y)),
    by.x = by, by.y = by) {
  stopifnot(
    is.data.frame(x),
    is.data.frame(y),
    length(by.x) == length(by.y))
  !do.call(paste, c(x[by.x], sep = "\30")) %in% do.call(paste, c(y[by.y], sep = "\30"))
}
# Example usage
# remove all 4 or 6 cylinder 4 gear cars or 8 cylinder 3 gear rows
to_remove <- data.frame(cyl = c(4, 6, 8), gear = c(4, 4, 3))
mtcars[setdiff.data.frame(mtcars, to_remove), ]
#>                 mpg cyl  disp  hp drat    wt  qsec vs am gear carb
#> Hornet 4 Drive 21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
#> Valiant        18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
#> Toyota Corona  21.5   4 120.1  97 3.70 2.465 20.01  1  0    3    1
#> Porsche 914-2  26.0   4 120.3  91 4.43 2.140 16.70  0  1    5    2
#> Lotus Europa   30.4   4  95.1 113 3.77 1.513 16.90  1  1    5    2
#> Ford Pantera L 15.8   8 351.0 264 4.22 3.170 14.50  0  1    5    4
#> Ferrari Dino   19.7   6 145.0 175 3.62 2.770 15.50  0  1    5    6
#> Maserati Bora  15.0   8 301.0 335 3.54 3.570 14.60  0  1    5    8
# with differing column names
to_remove2 <- data.frame(a = c(4, 6, 8), b = c(4, 4, 3))
mtcars[setdiff.data.frame(mtcars, to_remove2, by.x = c("cyl", "gear"), by.y = c("a", "b")), ]
#>                 mpg cyl  disp  hp drat    wt  qsec vs am gear carb
#> Hornet 4 Drive 21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
#> Valiant        18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
#> Toyota Corona  21.5   4 120.1  97 3.70 2.465 20.01  1  0    3    1
#> Porsche 914-2  26.0   4 120.3  91 4.43 2.140 16.70  0  1    5    2
#> Lotus Europa   30.4   4  95.1 113 3.77 1.513 16.90  1  1    5    2
#> Ford Pantera L 15.8   8 351.0 264 4.22 3.170 14.50  0  1    5    4
#> Ferrari Dino   19.7   6 145.0 175 3.62 2.770 15.50  0  1    5    6
#> Maserati Bora  15.0   8 301.0 335 3.54 3.570 14.60  0  1    5    8