Japanese 

                                                 1

There are couple of mnemonics for square root numbers :

sqrt(2) -> "hitoyo hitoyo ni hitomigoro"
                  1. 4  1 4     2 1   3  5 6

sqrt(3) ->  "hitonamini ogoreya"
                   1. 7 3 2  0 5 0 8

sqrt(5) ->  "fuji.sanroku oroya"  (??? I'm not sure)
                    2.2 3   6   0 6 8

From: Hajime Hirase

                                                 2

          1   4  1   4 2  1   3  5 6
sqrt(2) = hitoyo hitoyoni hitomi goro (worth seeing night after night)

          1   7 3 2  05 0 8
sqrt(3) = hitonamini ogoreya (pay for me like others do)

          2 2 3  6   2  0 6 7 9
sqrt(5) = fujisanrokuni oomunaku (a parrot sings in Mt. Fuji)

From: Naoto Takahashi

                                                 3
sqrt(2) = 1.41421356 
hito(1) yo(4) hito(1) yo(4) ni(2) hito(1) mi(3) go(5) ro(6)
"night by night, things seems to become better to be seen".

sqrt(3) = 1.7320508 
hito(1) na(7) mi(3) ni(2) o(0) go(5) re(0) ya(8)
"please buy me something like others".

sqrt(5) = 2.2360679 
fu(2) ji(2) san(3) roku(6) o(0) mu(6) na(7) ku(9)
"a parrot cries at the foot of Mt. Fuji".

sqrt(6) = 2.44949 
ni(2) yo(4) yo(4) ku(9) yo(4) ku(9)
"make it resemble to that very very much".

sqrt(8) = 2.8284 
ni(2) ya(8) ni(2) ya(8) shi(4)
"give a broad grin".

From: Kazuhiko Kudo