Ideally, one would always do ritual handwashing (netilas yadayim) and say Asher Yotzar after going to the bathroom. (Source: Shulchan Aruch OC 4:18 for handwashing; here for asher yatzar) However, in practice this can sometimes be hard to do--especially in public restrooms, and/or without a washing cup. Is it better in these cases to skip the blessing altogether, or to say it with (ritually) unwashed hands?
IS A CUP NEEDED? IS 3 TIMES PER HAND NEEDED?
Washing after the bathroom, doesn't require a utensil, M’Ikar D’Din, and even though it’s good to use one, when using a cup, the hands needn’t be washed more than once each hand, and they definitely don't require the “right, left, right, left” sequence that is required by Negel Vasser. (See Shulchan Aruch and Mishna Berura Siman 165:1. See also Mishnah Berurah Siman 4:39 and Aruch HaShulchan Siman 4:21. See also Shu”t Minchas Yitzchak Vol 5 Siman 96)
he Mishna Berura (Siman 4:39) mentions a stringent opinion that requires washing 3 times each hand after using the bathroom. However, he brings the Magen Avraham who argues with that opinion , and indeed it seems that the Mishna Berura concurs that it isn't necessary. See also Sha”arei Teshuva Siman 4:12 where he rules that no utensil, and no three times is necessary after the bathroom.
Moreover. the Elya Rabba Siman 4:12 writes that even the stringent opinions that require 3 times after the bathroom, agree that no utensil is necessary for those three times!
Another proof to this is brought by the Poskim from the language of the Shulchan Aruch Yoreh Deah Siman 376:4 where he requires washing the hands after being in a cemetery (which is the same Tumah status as after using the bathroom) where he writes “Rochtzim” and not “Notlim”, the language used for Netilas Yadayim, which would seem to mean him not requiring a utensil.
The only reason that the custom developed to use a utensil after the cemetery was due to the restriction on dipping the hands into the river. (See Gilyon MaHarsha on the above Shulchan Aruch and see Kitzur Shulchan Aruch Siman 199:10.)
Though many people are stringent to use a utensil it is a Chumra (stringency) and NOT M’Ikar D’Din (See Shu”t Minchas Yitzchok Vol. 5 Siman 96. See also Shu”t Yad HaLeivi where he rules that only after using a bathroom belonging to Aino Yehudims is a utensil required, but that is due to a danger and not M’Ikar D’Din. See also Biur haGra to Shulchan Aruch Siman 4:18 where he clearly states that this washing is not M’Ikar D’din and is due to the danger)
Bottom line, although there are indeed such opinions, most contemporary Poskim do not rule that way. They just say that as a Chumra, if one wants to do it he can, but they do not require it.
Many Talmidei Chachomim and Poskim are not stringent to use a utensil after using the bathroom.
Yalqut Yosef - Qizur Shulhan 'Arukh (Orah Hayim 3:2) states (my translation):
צריך ליטול ידיו כדי לברך אשר יצר... [אבל] מעיקר הדין מותר לומר דברים שבקדושה אחר ניקוי ידיו במידי דמנקי
One should wash one's hands in order to recite "Asher Yazar"... But, strictly according to the letter of the law, one is permitted to say devarim shebiqdushah after cleaning one's hands with midi dimnaqe'i (i.e. items that clean).
Therefore, later (Orah Hayim 6:14) he states (my translation):
בשעת הדחק [יכול] לנקות ידיו במידי דמנקי כדי לברך אשר יצר
In a pinch, one may clean one's hands with midi dimnaqe'i (i.e. items that clean) in order to recite "Asher Yazar".
In short, ascertain that your hands are clean of visible dirt, rub them gently along the wall (if running water is unavailable), and recite "Asher Yazar".