Meiri (Berachos 25b) and Rashba (Teshovos 1 153) imply that only prayers that include God's name are prohibited. See also language of Shulchan Aruch (OC 4 23), based on Rashba.
One is permitted to recite vidui in a bath house provided one doesn't mention the name of God. See Taz (OC 606 5) and Pri Megadim there, Kaf HaChaim (OC 74 28).
The gemara in Berachos (24b) requires you to wait for the odor to pass before reciting a prayer which does not mention the name of God. This seems to contradict the above. However it is possible that since it only involves waiting a short amount of time it is preferable to wait.
See also Biur Halacha (588) for a discussion of performing a mitzvah in a forbidden area, with astonishing conclusions.
I heard on a tape from Rabbi Belsky that one is permitted to be "mavdil" on motzei shabbos in the bathroom without employing prayer, ie. by saying "I acknowledge that the Sabbath has departed." This also implies that an actual prayer such as Baruch Hamavdil etc. should not be said.
See also Rosh Berachos (3 22) and Madanei Yom Tov there, which further implies that prayer without Gods name should not be said in any prohibited place.
Note that any form of Torah may not be said, for the whole of Torah is God's name (Aruch HaShulchan).