What generally accepted mnemonic phrases are there in Japanese that are used for memorizing a sequence of numbers or words?
I'm not sure about "generally accepted"; some are probably more widely known than others. I would not be surprised to observe patterns corresponding to the 流派 (schools) of traditional Japanese arts, where the goroawase students learn can be linked to geographical location/study guide used/target school/etc.
I am personally fond of the variants on "naku yo uguisu" in your example (e.g. "haku yo uguisu", "the warbler vomits" for 894, etc.), and of course "Iroiro sankyū, Porutogaru" for 1639 when Japan's ports were officially closed to Portuguese ships.
Square root of two
'It's peoples world; peoples world. It's time to observe people.'
Square root of three
'Treat me as if you do to other people.'
Square root of five
'Base of Mt. Fuji, a parrot cries.'
Base of natural logarithm
'One bowl, two bowls, one bowl, two bowls of crucian carp. Very delicious.'
Months that do not have 31 days
2, 4, 6, 9, 11
'A samurai facing west'
Beginning year of the Heian era
'Crying is a bush warbler, the Heian capital.'
Beginning year of the Kamakura era
'Let's make a good country: The Kamakura shogunate.'
Order of first elements in the periodic table
H (水素, すいそ), He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca
水兵リーベ僕の船 名前があるシップス クラークか （すいへいリーベぼくのふねなまえ…）
'Sailor, Liebe (de:love), my boat. Ships with names; maybe Clark?'
Chemical elements in the halogen group
F (フッ素; ふっそ), Cl (塩素; えんそ 'chlorine'), Br (臭素; しゅうそ 'bromine'), I (ヨウ素; ヨウそ 'iodine'), At (アンチモン 'antimony')
'A plumpy brasier: trace of love'