A query about na.omit

Dear all,
 
Say I have the following dataset:
 

DF

        x     y     z
[1]   1     1     1
[2]   2     2     2
[3]   3     3    NA
[4]   4   NA   4
[5]  NA  5     5
 
And I want to omit all the rows which have NA, but only in columns X and Y, so that I get:
 
 x  y  z
1  1  1
2  2  2
3  3  NA
 
If I use na.omit(DF), I would delete the row for which z=NA, obtaining thus
 
x y z
1 1 1
2 2 2
 
But this is not what I want, of course. 
If I use na.omit(DF[,1:2]), then I obtain
 
x y 
1 1
2 2
3 3
 
which is OK for x and y columns, but I wouldn't get the corresponding values for z (ie 1 2 NA)
 
Any suggestions about how to obtain the desired results efficiently (the actual dataset has millions of records and almost 50 columns, and I would apply the procedure on 12 of these columns)?
 
Sincerely,
 
Jose Luis 
 
Jose Luis Iparraguirre
Senior Research Economist 
Economic Research Institute of Northern Ireland